Rights  of  Translation  Reserved. 


THE  PROPAGATION  OF 
ELECTRIC  CURRENTS 

IN  TELEPHONE  AND  TELEGRAPH 
CONDUCTORS 


BY 

J.  A.   FLEMING,   M.A.,   D.Sc.,   F.R.S, 
• » 

FENDER  PROFESSOR  OF  ELECTRICAL  ENGINEERING  IN  THE  UNIVERSITY 
OF  LONDON;  MEMBER,  AND  PAST  VICE-PRESIDENT  OF  THE  INSTITUTION 
OF  ELECTRICAL  ENGINEERS;  MEMBER,  AND  PAST  VICE-PRESIDENT  OF  THE 
PHYSICAL  SOCIETY  OF  LONDON  ;  MEMBER  OF  THE  ROYAL  INSTITUTION 

OF    GREAT    BRITAIN,    ETC.,    ETC. 


NEW    YORK 

D.   VAN    NOSTRAND   COMPANY 

23    MURRAY   AND   27    WARREN   STREETS 

1911 


PBEFACE 


THIS  book  is  a  reproduction,  with  some  amplifications,  of  the 
notes  prepared  by  the  Author  for  two  Courses  of  Postgraduate 
Lectures  given  by  him  before  the  University  of  London  in 
the  Fender  Electrical  Laboratory  in  1910  and  1911,  on  the 
Propagation  of  Electric  Currents  in  Telephone  and  Telegraph 
Conductors  and  on  Electrical  Measurements  in  connection  with 
Telephonic  and  Telegraphic  work.  These  Lectures  had  their 
origin  in  a  request  made  to  the  University  to  provide  a  course 
of  instruction  for  Telegraphic  and  Telephonic  Engineers  which 
should  enable  them  to  keep  abreast  of  the  most  recent 
scientific  and  technical  researches  in  these  branches  of  Electrical 
Technology. 

These  Lectures  were  attended  by  a  large  class  composed  chiefly 
of  practical  Telegraphic  and  Telephonic  Engineers  and  experts ; 
and  at  the  request  of  many  who  attended,  and  some  who  did  not, 
the  Author  has  written  them  out  for  publication. 

As  a  considerable  portion  of  the  subject-matter  included  has  not 
yet  found  its  way  into  text-books,  although  distributed  through 
various  technical  Journals  and  Proceedings,  it  seemed  probable 
that  a  service  would  be  rendered  to  Electrical  Engineers  generally 
if  this  material  were  collected  and  placed  in  an  easily  accessible 
form.  Students  of  this  subject  are  well  aware  of  the  great  value 
of  the  pioneer  work  of  Mr.  Oliver  Heaviside  and  of  Prof.  Pupin 
in  laying  the  sound  theoretical  and  practical  foundations  for 
improvements  of  great  importance  in  telephony,  and  of  the 
classical  labours  of  Lord  Kelvin  in  connection  with  submarine 
telegraphy.  But  the  study  of  the  writings  of  these  originators 
makes  a  demand  for  mathematical  knowledge  which  is  generally 
beyond  the  attainments  of  the  practical  telegraphic  and  tele- 
phonic engineer.  Prof.  A.  E.  Kennelly  has  rendered  them, 

224489 


vi  PREFACE 

however,  an  immense  service  in  elaborating  mathematical 
methods  simple  in  character  and  capable  of  being  applied  in 
practical  calculations.  Much  of  Prof.  Kennelly's  instructive 
expositions  are,  however,  contained  in  periodicals  and  journals 
not  very  readily  obtained  by  British  telegraphists  or  readers. 

The  Author  has  accordingly  provided  in  the  first  place  a 
simple  mathematical  introduction  which  will  enable  any  technical 
student  to  acquire  easily  a  working  knowledge  of  the  mathe- 
matical operations  and  processes  required  in  conducting  the 
necessary  calculations  in  connection  with  this  subject.  In  the 
next  place  he  has  endeavoured  to  simplify  as  far  as  possible 
the  theoretical  treatment ;  and  thirdly,  by  illustrative  examples, 
to  render  it  possible  for  every  such  student  to  carry  out  readily 
the  arithmetic  calculations  by  means  of  hyperbolic  functions 
in  accordance  with  the  methods  which  have  been  admirably 
elucidated  by  Prof.  Kennelly  in  numerous  papers. 

The  Author  desires,  in  conclusion,  to  return  thanks  to  those 
who  have  assisted  or  furnished  information.  Major  O'Meara, 
C.M.G.,  Engineer-in-Chief  of  the  General  Post  Office,  has  most 
kindly  permitted  copious  extracts  and  the  loan  of  diagrams  from 
his  paper  read  in  1911  before  the  Institution  of  Electrical 
Engineers,  describing  the  Loaded  Anglo-Erench  Telephone  Cable 
laid  in  1910.  Mr.  F.  Gill,  M.Inst.E.E.,  Engineer-in-Chief  of  the 
National  Telephone  Company,  not  only  lent  apparatus  from  the 
investigation  laboratory  of  the  National  Telephone  Company  for 
illustrating  the  Lectures  as  given,  but  Has  kindly  furnished 
information  embodied  in  many  of  the  tables  in  this  book,  and 
also  permitted  special  measurements  to  be  made  in  his  research 
laboratory  by  Mr.  B.  S.  Cohen.  The  Author  desires  to  record 
his  particular  thanks  to  Prof.  A.  E.  Kennelly,  of  Harvard 
University,  for  permitting  a  free  use  to  be  made  of  all  his 
valuable  papers  and  writings  on  this  subject  and  the  appro- 
priation of  many  useful  tables  such  as  the  Tables  of  Hyperbolic 
Functions  of  Complex  Angles  in  Chapter  I.  and  the  Table  of 
Hyperbolic  Functions  in  the  Appendix.  Papers  published  by 
Messrs.  Cohen  and  Shepherd,  and  read  before  the  Institution 
of  Electrical  Engineers,  have  also  been  laid  under  contribution, 
and  to  them  an  acknowledgment  is  due.  Mr.  H.  Tinsley  also 


PREFACE  vii 

kindly  furnished  the  results  of  special  measurements  made  with 
artificial  cables,  and  also  granted  the  use  of  diagrams  of 
apparatus  made  by  his  firm.  The  Author  desires  also  to  include 
in  the  list  of  those  who  have  assisted  him,  Mr.  G.  B.  Dyke,  B.Sc., 
who  aided  him  efficiently  in  the  Lectures  by  taking  a  practical 
exercise  class,  and  has  also  made  or  checked  many  of  the 
calculations  and  assisted  in  reading  the  proofs  of  the  book.  In 
the  hope,  therefore,  that  these  republished  lectures  may  be 
useful  to  a  larger  number  of  telegraphists  and  telephonists  than 
those  to  whom  they  were  actually  delivered,  they  are  presented 
in  book  form,  and  may  serve  at  least  as  a  stepping  stone  or 
introduction  to  the  work  of  original  investigators  of  a  more 
advanced  or  difficult  character. 

J.  A.  F. 

UNIVERSITY  COLLEGE, 
LONDON, 
May,  1911. 


TABLE  OF  CONTENTS 


PAGE 

PREFACE  .  v 


CHAPTER  I 

MATHEMATICAL  INTRODUCTION 

1.  Introductory  ideas  and  definitions.  Statement  of  the 
problem  to  be  discussed.  Mean  square  value  of  a  periodic 
quantity.  Sine  curve  or  simple  harmonic  functions.  Ampli- 
tude and  phase  difference  of  simple  periodic  curves  or 
quantities.  Clock  diagrams.  —  2.  The  representation  of 
simple  periodic  quantities  or  vectors  by  complex  quantities. 
— 3.  The  calculus  of  complex  quantities.  Addition,  sub- 
traction, and  rotation  of  vectors.  Multiplication  and  division 
of  complex  quantities.  Rules  for  obtaining  the  size  of  a 
function  of  complex  quantities. — 4.  Hyperbolic  trigonometry. 
Relation  to  ordinary  or  circular  trigonometry.  Equation 
and  rectification  of  the  hyperbola.  Definition  and  tabulation 
of  hyperbolic  functions. — 5.  Formulas  of  hyperbolic  trigo- 
nometry. Graphic  representation  of  the  hyperbolic  functions 
of  complex  angles.  Construction  for  obtaining  a  vector 
representing  the  hyperbolic  sine  or  cosine  of  a  complex 
angle.  Dr.  A.  E.  Kennelly's  tables  of  the  hyperbolic 
functions  of  complex  angles.  Inverse  hyperbolic  functions 
and  mode  of  calculating  them. 


CHAPTER   II 

THE  PROPAGATION  OF  ELECTROMAGNETIC  WAVES  ALONG  WIRES  .      43 

1.  Wave  motion.  Qualities  of  a  medium  in  which  wave 
motion  can  exist.  Theory  of  longitudinal  wave  motion. 
Formula  for  wave  velocity  in  a  gas. — 2.  The  electromagnetic 
medium.  Its  properties.  The  electron.  Electric  strain  and 


TABLE  OF  CONTENTS 


displacement.  Lines  of  electric  strain  or  force.  The  nature 
of  an  electron  or  strain  centre. — 3.  Electric  and  magnetic 
forces  and  fluxes.  The  properties  of  lines  of  electric  strain. 
The  magnetic  effect  of  a  moving  electric  charge.  Rowland's 
experiment.  The  reciprocal  relation  of  moving  lines  of 
electric  strain  and  magnetic  flux.  The  curl  of  a  vector.  The 
relations  between  the  curls  of  the  magnetic  and  electric 
forces. — 4.  Electromagnetic  waves  along  wires  :  their  nature 
and  motion. — 5.  The  reflection  of  electromagnetic  waves  at 
the  end  of  a  line  when  open  or  short-circuited. — G.  The 
differential  equations  expressing  the  propagation  of  an 
electromagnetic  disturbance  along  a  pair  of  wires.  Definition 
of  the  vector  impedance  and  admittance.  The  propagation 
constant  of  a  line.  The  primary  constants  of  a  line.  The 
attenuation  and  wave  length  constants  of  a  line.  Formulas 
for  the  same  in  certain  reduced  cases.  Definition  of  a 
distorsionless  cable. 


CHAPTER   III 

THE  PROPAGATION  OF  SIMPLE  PEEIODIC  ELECTRIC  CURRENTS 
IN  TELEPHONE  CABLES 

1.  The  case  of  an  infinitely  long  cable  with  simple  periodic 
electromotive  force  applied  at  the  sending  end.  The 
differential  equations  for  propagation  and  their  solution. 
The  initial  sending  end  impedance  of  the  line.  The 
attenuation  factor  and  phase  factor.  A  model  representing 
the  variation  of  current  and  potential  at  various  points  in  a 
telephone  line  subjected  to  a  simple  periodic  electromotive 
force  at  the  sending  end.  The  variation  of  wave  velocity 
and  of  attenuation  with  frequency. — 2.  The  propagation  of 
simple  periodic  currents  along  a  line  of  finite  length.  Solu- 
tion of  the  differential  equations  for  this  case. — 3.  The 
propagation  of  currents  along  a  finite  cable  free  or  insulated 
at  the  receiving  end.  Solution  of  the  differential  equations 
for  this  case.  Initial  and  final  sending  end  and  receiving 
end  impedance.  The  effect  of  reflections  at  the  end.  The 
hyperbolic  functions  which  express  the  summation  of  these 
reflections. — 4.  Propagation  of  current  along  a  line  short- 
circuited  at  the  receiving  end. — 5.  The  propagation  of  simple 
periodic  currents  along  a  transmission  line  having  a  receiving 
instrument  of  known  impedance  at  the  end.  The  solution 
of  the  differential  equations  for  this  case.  Abbreviated 
formulas  for  the  impedances  and  ratio  of  sending  end  to 
receiving  end  currents. 


TABLE    OF   CONTENTS  xi 

CHAPTER   IV 

PACE 

TELEPHONY  AND  TELEPHONIC  CABLES 90 

1.  The  principles  of  telephony.  General  nature  of  a  tele- 
phonic circuit  and  apparatus.  The  amplitude  of  sound 
waves.  The  wave  form  of  sound  waves.  Vowel  and  con- 
sonantal sounds. — 2.  Fourier's  theorem.  The  analysis  of 
complex  single-valued  curves  into  the  sum  of  a  number  of 
sine  curves  differing  in  amplitude  and  phase,  The  analytical 
proof  of  Fourier's  theorem.  Mode  of  finding  the  constants. 
A  numerical  example  of  the  harmonic  analysis  of  a  complex 
curve. — 3.  The  analysis  and  synthesis  of  sounds.  Von 
Helmholtz's  experiments  on  vowel  sounds.  The  quality  of 
sounds. — 4.  The  reasons  for  the  limitations  of  telephony. 
The  distorsional  qualities  of  the  line. — 5.  The  improvement 
of  practical  telephony.  Mr.  Oliver  Heaviside's  suggestions. 
His  distorsionless  cable.  His  proposed  remedies  for  dis- 
torsion.  Prof.  Pupin's  work  and  papers.  Pupin's  suggestion 
for  a  loaded  line. — 6.  Pupin's  analytical  theory  of  the 
unloaded  line. — 7.  Pupin's  theory  of  the  loaded  cable. 
Pupin's  rule  for  spacing  the  loading  coils. — 8.  Campbell's 
theory  of  the  loaded  cable.  Calculation  of  the  average 
attenuation  constant  of  a  loaded  line. — 9.  Other  proposed 
methods  of  reducing  line  distorsion. — 10.  The  theory  of  the 
S.  P.  Thompson  cable  with  inductive  shunts.  Roeber's 
investigation  of  the  same. — 11.  Other  forms  of  distorsionless 
cable  proposed  by  Prof.  S.  P.  Thompson  and  Reed. 

CHAPTER  V 
THE  PEOPAGATION  OF  CURRENTS  IN  SUBMARINE  CABLES  .        .    142 

1.  The  differential  equation  expressing  the  propagation  of  a 
current  in  a  cable. — 2.  The  reduced  case  applicable  in  sub- 
marine telegraphy.  The  telegraphic  equation.  Lord  Kelvin's 
classical  investigations  in  1855. — 3.  The  theory  of  the  sub- 
marine cable.  Analysis  of  the  effect  of  applying  at  one  end 
a  brief  electromotive  force. — 4.  Curves  of  arrival  and  mode 
of  predetermining  them.  Calculation  of  the  currents  arriving 
at  the  receiving  end  of  a  cable. — 5.  The  transmission  of 
telegraphic  signals.  The  syphon  recorder.  A  simple  dot 
and  dash  signal.  Graphical  representation  of  the  same  as 
sent  and  received.  Mode  of  predetermining  the  form  of  a 
received  signal  for  any  letter  sent  along  a  submarine  cable. 
—6.  Speed  of  signalling.  Rules  for  calculating  it. — 7.  Curb 
sending.  A  curbed  signal.  Duplex  transmission.  The 
usual  form  of  apparatus  for  duplex  cable  signalling.  The 
form  of  the  received  signals  as  affected  by  the  length  and 
constants  of  the  cable. 


xii  TABLE   OF   CONTENTS 

CHAPTER  YI 

PAGE 

THE  TRANSMISSION  OF  HIGH  FREQUENCY  AND  VERY  Low 

FREQUENCY  CURRENTS  ALONG  WIRES  .   .   .    .171 

1.  The  modification  in  the  general  differential  equation  for 
transmission  in  the  case  of  very  high  and  very  low  frequency 
currents. — 2,  The  propagation  of  high  frequency  currents 
along  wires. — 3.  Stationary  oscillations  on  wires  of  finite 
length  when  subjected  to  a  simple  periodic  electromotive 
force  at  one  end. — 4.  The  production  of  loops  and  nodes  of 
potential  on  a  conductor  by  high  frequency  electromotive 
force.  Calculation  of  the  velocity  of  propagation  for  a 
certain  case.  Experimental  confirmation  of  theory  and 
description  of  apparatus  used  for  the  visible  production  of 
stationary  electric  oscillations  on  helices  of  wire.  The 
author's  experiments  with  helices. — 5.  The  propagation  of 
currents  along  leaky  lines.  The  modification  in  the  general 
differential  equation  necessary  to  meet  this  case.  Application 
in  the  case  of  continuous  currents  in  leaky  lines. 


CHAPTER  VII 

ELECTRICAL    MEASUREMENTS    AND    DETERMINATION    OF    THE 

CONSTANTS  OF  CABLES 187 

1.  The  necessity  for  the  accumulation  of  data  by  practical 
measurements. — 2.  The  predetermination  of  the  capacity  of 
conductors  for  certain  cases  such  as  spheres  and  wires. — 
3.  The  capacity  of  overhead  telegraph  wires.  Formula  for 
the  same. — 4.  The  capacity  of  concentric  cylinders  and  of  a 
submarine  cable. — 5.  Formulae  for  the  inductance  of  cables. 
Case  of  two  parallel  wires.  Neumann's  formula  for  the 
mutual  inductance  of  two  circuits.  The  mutual  inductance 
of  a  pair  of  parallel  wires.  Definition  of  geometric  mean 
distance. — 6.  The  practical  measurement  of  the  capacity  of 
telegraph  and  telephone  cables.  The  measurement  of  the 
capacity  of  a  leaky  condenser  or  one  with  conductive 
dielectric.  Dr.  Sumpner's  wattmeter.  Use  in  measuring 
capacity. — 7.  The  practical  measurement  of  inductance. 
Anderson-Fleming  method  for  measuring  small  inductances. 
— 8.  The  measurement  of  small  alternating  and  direct 
currents.  The  Duddell  therrnogalvanometer.  The  Cohen 
barretter. — 9.  The  measurement  of  small  alternating  voltages. 
The  alternate  current  potentiometer.  The  Drysdale  phase 
shifting  transformer.  The  Drysdale  -  Tinsley  alternate 
current  potentiometer.  The  Tinsley  vibration  galvanometer. 
The  method  of  using  the  potentiometer  to  measure  the  phase 


TABLE   OF   CONTENTS  xiii 


difference   and    strength    of   small  alternating  currents. — 

10.  The  measurement  of  the  attenuation  constant  of  cables. — 

11.  The  measurement  of  the  wave  length  constant  of  cables. 
—12.   The   measurement  of   the  propagation   constant   of 

cables. — 13.  The  measurement  of  the  initial  sending  end 
impedance  of  cables. — 14.  The  measurements  of  the  impe- 
dance of  various  receiving  instruments.  The  use  of  the 
Cohen  barretter  for  this  purpose.  Table  of  the  impedances 
of  various  pieces  of  telephonic  apparatus. — 15.  The  power 
absorption  of  various  telephonic  instruments.  — 16.  The 
determination  of  the  fundamental  constants  of  a  cable  from 
measurements  of  the  final  sending  end  impedance.  Results 
of  Messrs.  Cohen  and  Shepherd. 


CHAPTER  VIII 

CABLE    CALCULATIONS    AND    COMPARISON    OF    THEORY    WITH 

EXPERIMENT    ..........     233 

1.  Necessity  for  the  verification  of  formulae. — 2.  The  calcu- 
lation of  the  current  at  any  point  in  a  cable  either  earthed 
or  short-circuited  at  the  far  end  when  a  simple  periodic 
electromotive  force  is  applied  at  the  sending  end.  Com- 
parison of  the  formula  with  results  of  actual  measurements 
made  with  the  Drysdale-Tinsley  potentiometer  on  an  artificial 
cable. — 3.  Calculation  of  the  current  at  any  point  in  a  cable 
having  a  receiving  instrument  of  known  impedance  at  the 
far  end.  Comparison  of  the  formula  with  the  results  of 
actual  measurement  made  with  an  artificial  cable. — 4.  The 
calculation  of  the  voltage  at  the  receiving  end  of  a  cable 
when  open  or  insulated  and  of  the  current  when  closed  or 
short-circuited. — 5.  The  calculation  and  predetermination  of 
attenuation  constants.  Tabulated  results.  The  attenuation 
constant  of  loaded  cables.  The  attenuation  constant  of  the 
Anglo-French  loaded  telephone  cable. — 6.  Tables  and  data 
for  assisting  cable  calculations.  Tables  of  data  of  various 
sizes  of  telephone  cables  and  wires  obtained  used  by  the 
National  Telephone  Company. 


CHAPTER  IX 
LOADED  CABLES  IN  PRACTICE 2t>3 

1.  Modern  improvements  in  telephone  cables.  Uniform  and 
non-uniform  loading.  Effect  of  loading  on  aerial  lines. 
Necessary  qualities  of  telephonic  speech. — 2.  The  intro- 
duction of  loading  coils  into  overhead  or  aerial  lines.  Early 


xiv  TABLE   OF   CONTENTS 

experiments  in  Germany  on  the  Berlin-Magdeburg  line. 
The  effect  of  loading  on  the  attenuation  lengths  of  cables. 
The  limits  of  telephonic  speech.  Necessity  for  terminal 
taper.  Experiments  by  Dr.  Hammond  Y.  Hayes. — 3.  Loaded 
underground  cables.  Effect  of  terminal  taper  on  the 
attenuation.  Importance  of  maintaining  good  insulation  on 
cables.  Data  for  loading  coils  used  on  telephonic  cable 
circuits. — 4.  Loaded  submarine  telephone  cables.  Experi- 
ments at  the  General  Post  Office  on  G.  P.  wire.  Data  for 
some  uniformly  loaded  Danish  cables.  Some  foreign  con- 
tinuously loaded  cables.  The  loaded  telephone  cable  laid  in 
Lake  Constance.  The  British  Post  Office  loaded  Anglo- 
French  telephone  cable  of  1910.  Specification  for  its 
manufacture.  Its  manufacture  and  laying  by  Messrs. 
Siemens  Bros.  Constants  of  this  cable  given  by  Major 
O'Meara,  and  tests  of  the  cable. — 5.  The  effect  of  dielectric 
leakage  on  the  attenuation  constant  of  a  loaded  cable.  Dr. 
Kennelly's  researches.  Theory  of  the  leaky  loaded  cable. 
Some  data  from  telephone  cables  obtained  at  the  General 
Post  Office.  The  difficulties  of  loading  aerial  lines. 


THE  PROPAGATION  OF  ELECTEIC 

CURRENTS  IN  TELEPHONE  AND 

TELEGRAPH  CONDUCTORS 

CHAPTEK   I 

MATHEMATICAL    INTRODUCTION 

1.  Introductory    Ideas    and     Definitions.  --  The 

object  of  these  lectures  is  to  discuss  in  as  simple  a  manner 
as  possible  the  phenomena  connected  with  the  propagation 
of  electric  currents  in  telephone  and  telegraph  conductors. 
This  discussion  is  intended  to  provide  telegraph  and  telephone 
engineers  with  some  necessary  information  to  enable  them  to 
follow  the  original  writings  of  leading  investigators,  and  also 
with  the  means  of  solving  for  themselves  practical  problems 
in  connection  with  the  subject. 

Broadly  speaking,  the  chief  scientific  problem  which  presents 
itself  for  solution  in  connection  with  this  matter  is  that  of 
calculating  the  current  at  any  time  and  place  in  a  linear  con- 
ductor of  length  very  great  in  comparison  with  its  diameter, 
when  an  electromotive  force  of  known  type  and  magnitude  is 
applied  at  some  point  in  it.  Associated  with  this  is  the 
investigation  of  the  effects  produced  by  varying  the  nature  of 
the  conductor  and  of  the  terminal  apparatus  upon  the  current 
so  transmitted. 

The  conductors  we  shall  consider  may  be  either  bare  over- 
head wires,  underground  or  submarine  cables,  or  telephone 
wires  or  cables  of  different  kinds.  These  conductors,  in  any  case, 
have  four  specific  qualities  which  may  be  reckoned  per  unit  of 
length,  say  per  mile  or  per  kilometre. 

B.C.  13 


2     PROPAGATION.  OF:  ELECTRIC  CURKENTS 

These  qualities  are  — 

(i.)  The  resistance  of  the  conductor  per  unit  of  length  (It). 

(ii.)  The  inductance  of  the  conductor  per  unit  of  length  (L). 

(iii.)  The  electrical  capacity  per  unit  of  length  taken  with 
reference  to  the  earth  or  some  other  conductor  (C). 

(iv.)  The  insulation  resistance  of  the  dielectric  surrounding  the 
conductor  per  unit  of  length,  or  its  reciprocal  the  insulation 
conductivity  ($). 

The  above  quantities  are  all  of  the  type  called  scalar,  that  is 
they  are  completely  denned  as  to  amount  by  reference  to  a  unit 
of  the  same  kind. 

It  is  usual  to  reckon  the  resistance  in  ohms  per  mile  or 
kilometre,  the  inductance  in  henrys  or  millihenrys  per  mile  or 
kilometre,  the  capacity  in  microfarads  per  mile  or  kilometre, 
and  the  insulation  resistance  in  megohms  per  mile  or  kilometre, 
or  conversely  the  insulation  conductance  in  the  reciprocal  of 
megohms  per  mile  or  kilometre,  viz.,  in  mhos  per  mile  or 
kilometre.  We  have  then  to  consider  the  current  and  electro- 
motive force  at  any  point  in  the  conductor.  We  may  specify 
either  their  instantaneous  values,  that  is  the  value  they  have 
at  any  instant,  or  if  they  vary  cyclically  we  may  specify 
some  function  of  their  instantaneous  values  throughout  the 
period. 

The  instantaneous  value  of  the  current  at  any  point  in  the 
conductor  is  measured  by  the  ratio  of  the  quantity  of  electricity 
dq  which  flows  across  the  section  of  the  conductor  at  that  point 
in  any  time  dt  to  that  interval  of  time,  when  the  interval  is  taken 
exceedingly  small.  If  i  denotes  the  current  at  any  instant  and 
dq  the  quantity  of  electricity  which  flows  past  any  section  of  the 
conductor  in  the  time  dt,  then  we  have 


The  letter  q  with  a  dot  over  it  signifies  the  time  rate  of  change 
of  q.  If,  however,  the  current  varies  in  any  manner,  but  so 
that  it  passes  through  a  cycle  of  values  in  the  time  T,  called  the 
periodic  time,  then  the  insertion  of  a  hot  wire  ammeter  in  the 
circuit  at  that  point  will  give  us  a  reading  which  is  proportional 
to  the  square  root  of  the  mean  of  the  squares  of  the  instantaneous 


MATHEMATICAL   INTRODUCTION  3 

values  of  the  current  taken  at  small  and  numerous  equidistant 
intervals  of  time. 

This  function  of  the  instantaneous  values  is  called  the  root- 
mean-square  value  or  the  R.M.S.  value  of  the  current. 
Mathematically  it  is  expressed  by  the  equation 


B.M.S.  value  of  i  = 


(It 


(2) 


As  a  rule  we  are  not  much  concerned  with  the  true  arithmetic 
mean  value  of  the  instantaneous  current  throughout  a  period. 


\ 


270 


360 


180\ 


\ 


FIG.  1. — A  Sine  Curve. 


When,  however,  we  do  have  to  mention  it,  it  will  be  denoted  by 
the  symbols  T.M.  value  of  i  which  is  otherwise  expressed 


T.M.  value  of  i  =  -^ 


(It 


(3) 


In  a  large  number  of  problems  the  current  either  varies  or 
can  be  assumed  to  vary  as  the  ordinates  of  a  simple  curve  of 
sines. 

Take  any  straight  line  to  represent  the  periodic  time  and 
divide  it  say  into  24  parts.  At  successive  points  set  up  lines 
proportional  in  length  to  Sin  0°,  Sin  15°,  Sin  30°,  etc.  Join  the 
top  of  these  lines  by  a  smooth  curve  and  we  have  the  curve 
called  a  aim-  cur  re  (see  Fig.  1).  In  this  way  two  or  more  sine 
curves  may  be  drawn  differing  in  amplitude  or  maximum  value 
and  in  jthaw  or  zero  point  (see  Fig.  2). 

Taking  the  point  on  the  left  hand  at  which  the  ordinate  has 
its  zero  value  we  can  reckon  the  abscissa  of  any  point  on  the 
curve  as  equal  to  an  interval  of  time  t  on  the  same  scale  that  the 

B  2 


4  PEOPAGATION   OF   ELECTKIC  CUKEENTS 

whole  period  is  equal  to  T.     Hence  this  abscissa  reckoned  as  an 
angle  in  circular  measure  is  denoted  by  2ir  ^  the  periodic  time 

being  denoted  as  an  angle  by  2/r.     It  is  usual  to  write  p  for  ^,  and 

hence   the  abscissa  of   any   point   on   the   sine  curve  may   be 
represented  by  pt  in  angular  measure. 

If  the  ordinate  is  denoted  by  i  and  the  maximum  ordinate  by 
I  we  have  then  the  equation  to  the  sine  curve  in  the  form 

i  =  ISmpt (4) 

If  the  origin  from  which  we  reckon  our  time  is  not  the  zero 
point  of  the  curve,  but  some  point  more  to  the  left  of  it,  such  as 


FIG.  2. — Sine  Curves  differing  in  phaSe. 

the  point  0  in  Fig.  2,  then  the  equation  to  the  two  curves  in 
that  diagram  may  be  written 

*!==/!  Sin(jp*-<k) 
4  =  I2  Sin  (pt-fa) 

The  angles  <pi  and  c/>2  are  called  the  phase  angles  of  the 
zero  point  and  the  angle  </>i— <jf>2  is  called  the  difference  of  phase 
of  the  curves. 

It  is  clear,  therefore,  that  to  fix  the  position  and  form  of 
these  curves  we  require  to  know  two  parameters  for  each,  viz., 
the  maximum  value  I  and  the  phase  angle  $  relative  to  some 
point. 

We  can  represent  the  curve  in  another  manner. 

Suppose  a  line  OP  of  length  equal  to  the  maximum  value  I 
to  revolve  round  one  extremity  like  the  hand  of  a  clock  but  in 
a  counter-clockwise  direction  (see  Fig.  3).  Then  if  we  reckon 


MATHEMATICAL   INTRODUCTION  5 

angles  from  a  fixed  line  OQ  so  that  QOM  =  </>  and  QOP  =  pt 
and  hence  MOP  =  pt  —  </>,  it  is  clear  that  the  projection  of 
OP  on  the  vertical  OY,  viz.,  Op,  is  equal  to 

OP  Sin  (pt-<l>)  =  ISm  (pt-<j>)  =  i. 

Accordingly    the     magnitude    of    the    projection    Op  which 
represents    the  instantaneous  value  of   the  current  or  electro- 


FIG.  3.— Clock  Diagram. 

motive  force  is  determined  by  the  length  of  the  line  OP  and  its 
slope  at  the  corresponding  instant. 

Hence  an  alternating  or  simple  periodic  current  which  varies 
from  instant  to  instant  proportionately  to  the  ordinates  of  a 
sine  curve  can  he  represented  by  a  radial  line  drawn  in  a  certain 
position  on  a  clock  diagram  as  above  described. 

It  can  easily  be  shown  that  the  mean  value  of  Sin2  6  taken  at 
equidistant  numerous  intervals  of  the  angle  6  throughout  a  period 
or  between  0=0°  and  6  =  360°  is  equal  to  L 

For 


6  PKOPAGATION   OF   ELECTKIC   CUERENTS 

Now  the  mean  value  of  Sin  6  or  Cos  6  throughout  one  period 
or  from  0°  to  860°  is  zero ;  because  for  every  positive  value  of 
the  ordinate  of  the  curve  representing  these  functions  there  is  an 
equal  negative  value.  Therefore  the  mean  value  of  |  Cos  2  6 
throughout  a  period  or  from  6  =  0°  to  6  =  860°  is  zero,  and 
therefore  the  mean  value  of  Sin2  6  is  ^.  Therefore  the  root- 

mean-square  value  of  the  ordinate  of  a  sine  curve  is  -/=  where  / 

is  the  maximum  value.  In  a  clock  diagram,  therefore,  if  the 
revolving  radii  represent  maximum  values  of  the  currents  or 
E.M.F.,  dividing  them  by  V2  gives  the  li.M.S.  values,  assuming 
that  they  follow  a  simple  sine  law. 

We  shall  see  later  on  that  any  wave  form  may  be  resolved  into 
the  sum  of  a  number  of  sine  and  cosine  curves,  and  that  therefore 
certain  propositions  which  are  true  of  sine  curves  are  true  also  of 
periodic  curves  of  any  kind. 

For  the  present,  however,  we  may  limit  ourselves  to  the  con- 
sideration of  simple  periodic  electric  currents  represented  by  a 
simple  sine  curve. 

2.  The  Representation  of  Simple  Periodic 
Currents  by  Complex  Quantities.— Having  seen  that 
a  simple  periodic  current  may  be  represented  by  the  projection 
of  a  revolving  radius  on  a  diametral  line  through  the  centre 
of  revolution,  we  have  next  to  consider  how  such  a  line  can  be 
algebraically  specified. 

Suppose  we  draw  two  lines  at  right  angles  through  any  point, 
one  horizontal  and  one  vertical,  we  can  with  the  usual  conven- 
tions as  to  signs  represent  by  +  a  anJ  horizontal  line  a  units 
in  length  drawn  to  the  right  starting  from  the  origin.  Also  by 
—  a  any  horizontal  line  drawn  to  the  left. 

How  then  shall  we  represent  a  line  a  units  in  length  drawn 
vertically  through  the  origin  upwards  or  downwards  ?  We  can 
do  this  by  making  use  of  some  symbol  which  shall  denote  that 
the  horizontal  line  +  a  is  turned  through  a  right  angle  round  its 
left  extremity  in  a  counter-clockwise  or  clockwise  direction. 
This  symbol  must  be  such  that  when  prefixed  to  the  symbol  a 
it  denotes  a  line  drawn  vertically  upwards  through  the  origin. 


MATHEMATICAL   INTRODUCTION  7 

Also  it  must  be  such  a  symbol  tbat  when  twice  repeated  it  con- 
verts +  a  into  —  a>  since  turning  the  horizontal  line  through  two 
right  angles  reverses  its  direction.  LetJ  be  this  symbol.  Then 
ja  is  to  signify  a  line  of  a  units  in  length  drawn  vertically 
upwards  through  the  origin  or  the  line  a  turned  through  one 
right  angle.  Hence  jja  or  j*a  must  signify  a  horizontal  line  +  a 


-CL 


+J(L 


-J0- 


FIG.  4. 

turned  through  two  right  angles  or  reversed  in  direction.  There- 
fore, fa  =  —  a,  and  hence  j  —  V^l. 

The  symbol  j  therefore  considered  as  an  operator  or  sign  of  an 
operation  is  equivalent  in  meaning  to  V  —  1. 

We  have  then  the  following  symbols.  A  line  of  a  units  in 
length  drawn  horizontally  from  an  origin  is  denoted  by  +  a,  a 
line  of  the  same  length  drawn  vertically  upwards  is  denoted 
by  ja,  a  line  of  the  same  length  drawn  to  the  left  is  -  a,  and  an 
equal  line  drawn  vertically  downwards  is  —  ja  (see  Fig.  4). 

If  then  we  give  to  the  sign  of  addition  ( -h )  an  extended  meaning 


8 


PEOPAGATION  OF  ELECTEIC  CUERENTS 


to  make  it  signify  joint  effect,  we  can  say  that  the  expression 
a  -\-jb  signifies  a  straight  line  drawn  from  any  point  in  such  a 
direction  that  its  horizontal  projection  is  a  and  its  vertical 
projection  is  b  (see  Fig.  5). 

For  the  expression  a-\-jb  instructs  us  to  measure  a  length  a 
starting  from  the  origin  in  a  horizontal  direction.  Then  to 
measure  off  a  length  b  in  a  vertical  position  starting  from  the 
end  of  a,  and  the  joint  effect  of  these  two  steps  is  the  same  as  if 
we  had  moved  over  a  straight  line  of  length  \/a?  +  b~  inclined  at 

an  angle  6  to  the  horizontal  such  that  tan  6  =  -.  The  quantity 
a-\-jb  equivalent  to  a  +  V  —  1  b  is  called  a  complex  quantity, 


+  a 
FIG.  5. 

is  called  its  modulus 


or  size,  and  0  =  tan  *  -  its 

Cb 


and  Va2  + 

slope. 

The  part  a  is  called  the  horizontal  step  and  b  is  called  the  vertical 
step.  Hence,  a  -\-jb  stands  for  a  straight  line  or  anything  which 
has  magnitude  and  direction,  such  as  a  force,  velocity,  or  accelera- 
tion. In  other  words,  a-\-jb  stands  for  a  vector  quantity; 
whilst  Va2  +  ^2  denotes  its  size,  or  mere  magnitude  apart  from 
direction.  We  shall  in  future,  following  a  common  custom, 
denote  vectors  considered  as  vectors  by  letters  printed  in  thick 
or  Clarendon  type.  Thus  A  signifies  a  vector  or  stands  for 
a  -{-jb.  We  shall  denote  the  mere  size  or  modulus  by  an  ordinary 
Eoman  capital.  Thus  A  stands  for  Va2  +  b2.  It  is  more  con- 
venient sometimes  to  denote  the  mere  size  or  length  of  a  vector  A 


MATHEMATICAL   INTRODUCTION 


We 


by  brackets,  e.a.  (A).  The  student  should  note  that  a  +  jb 
signifies  not  merely  a  line  drawn  from  one  origin,  but  any  line  of 
the  same  length  and  with  the  same  slope  drawn  from  any  point 
in  the  same  direction. 

We  have  seen  that  a  simple  periodic  or  alternating  electro- 
motive force  or  current  can  be  represented  by  a  radial  straight 
line  the  length  of  which  is  proportional  to  the  maximum  value 
of  or  amplitude  of  the  periodic  quantity  and  its  slope  to  the 
phase  with  respect  to  some  instant  of  time.  Accordingly  such  a 
simple  periodic  current  or  E.M.F.  can  be  denoted  by  a  complex 
quantity  such  as  a  -j-  jb.  The  amplitude  of  the  quantity  will 

be  measured  by  Va2  -f-  b1  and  its  R.M.S.  value  by 

have  then  to  consider  the  rules  for  handling  complex  quantities 
in  calculations. 

3.  The    Calculus    of   Complex    Quantities.— Let 

A  =  a  -\-jb  and  B  =  c  -f  jd  be  two  complex  quantities  or 
vectors ;  then  if  A  =  B  it 
signifies  that  the  vectors  or 
lines  representing  them  are 
equal  and  parallel.  Accord- 
ingly, if  we  draw  these  lines 
and  set  off  their  horizontal 
and  vertical  steps  (see  Fig.  6), 
it  is  clear  that  the  triangles 
so  formed  are  similar  and 
the  side,  A  is  equal  to  the 
side  B.  Hence  we  have  also 
a  =  c  and  b  =  d.  In  other 
words,  if  two  complexes  are 
equal  we  may  equate  the 
horizontal  and  vertical  steps 
respectively. 

In  the  next  place  let   us 
consider  the  result  of  adding 

together  two  complexes.  In  this  process  addition  is  equivalent 
to  joint  effect.  The  complexes  represent  lines  and  must  be 
added,  therefore,  like  forces,  by  the  parallelogram  law. 


CL 


c 

FIG.  6. 


10 


PROPAGATION  OF  ELECTRIC  CURRENTS 


If  a  -\-  jb  and  c  +  jd  are  two  complexes  representing  lines 
OA,  OB  drawn  from  the  origin,  then  their  resultant  or  vector 
sum  is  OD,  the  diagonal  of  the  parallelogram  formed  on  them, 
It  is  clear,  therefore,  from  Fig.  7  that  OD  is  a  vector  whose 
horizontal  step  is  a  +  c  and  vertical  step  b  +  d.  Hence 

a+jb  +  c+jd  =  a+c+j  (b+d). 
The  second  rule  is  then 

To  add  together  two  complexes,   add   the   respective    horizontal 


a  c 

FIG.  7. — Addition  of  Vectors. 

steps  for  the  resultant  horizontal  step,  and  the  respective  veitical 
steps  for  the  resultant  vertical  step. 

Ex.—Md  together  5  +  j6  and  7  +  j  9.     Am.  12  +j  15. 

The  same  process  may  be  extended  to  any  number  of  com- 
plexes. If  ai+j'&i,  az-\-jb%,  etc.,  are  several  vectors,  then 
their  vector  sum  is  2&  +  feb,  where  2a  stands  for  the  algebraic 
sum  of  all  the  horizontal  steps  and  2b  of  all  the  vertical  steps. 
It  follows  that,  if  the  vector  sum  is  zero  and  if  the  lines  be  taken 
to  represent  forces,  these  forces  are  in  equilibrium  ;  also  that 
the  sides  of  a  polygon  taken  in  order  are  parallel  and  propor- 
tional to  these  forces  in  equilibrium. 


MATHEMATICAL   INTRODUCTION  11 

Example.—  (jive,  expressions  in  complex  form  for  the  sides  of 
a  hexagon. 

Ans.  —  Let    one  side  be   horizontal   and    of    length  a.      The 

next  side  is  represented  by  |+y  -^  a,  the  third  by  —  |+y  -^  a, 
the    fourth   by  —  a,  the  fifth  by  —  TJ—  /-a"  a,  and  the  sixth  by 

9  —j  —n  a.    The  vector  sum  is  zero.     Hence  forces  parallel  and 

proportional  to  the  sides  of  a   hexagon  taken  in  order   are  in 
equilibrium. 

As  a  preliminary  to  additional  propositions  we  must  exhibit 
other  expressions  for  complex  quantities.  If  a  -\-  jb  is  a  complex 
and  6  its  slope,  then  obviously  a  =  A  Cos  0  and  b  =  A  Sin  6. 
Hence  we  have 

a+jb  =  A  =  &  (Cos  0+j  Sin  0). 

The  quantity  A  is  the  size  of  the  vector  or  is  Va2  +  Ir.  The 
quantity  (Cos  6  +  j  Sin  0)  is  called  a  rotating  operator  or  rotator. 
The  effect  of  it  when  applied  to  a  vector  quantity  is  to  turn  the 
vector  through  an  angle  6  without  altering  its  size.  Thus 
Va2  +  b2  represents  a  length  or  line  set  off  in  a  horizontal  direc- 
tion ;  but  Va2  +  b2  (Cos  6  -\-  j  Sin  0)  is  a  line  of  the  same  length 
making  an  angle  0  with  the  horizontal.  Hence  any  expression 
of  the  form  A  (Cos  6  +  j  Sin  6)  represents  a  line  of  length  A 
and  slope  6. 

We  can  easily  prove  that  the  modulus  or  size  of  the  complex 
quantity  (a  -\-jb)  (Cos  9  -\-j  Sin  6)  is  the  same  as  the  modulus 
of  a  -\-  jb,  viz.  Va2  +  b2,  but  the  slope  of  the  former  vector  is 
greater  than  that  of  the  latter  by  an  angle  6. 

For   (a  +jb)    (Cos   0  +  ./   Sin   0)  =  (a   Cos   0  -  b   Sin  0) 
+  j  (b  Cos  0  +  a  Sin  0). 

Now  the  size  of  the  latter  complex  is 

•J  (a  Cos  0  —  b  Sin  0)*  +  (b  Cos  0+a  Sin  fff  =  \/a2  +  62 


and  the  slope  of  this  vector  is  an  angle  (f)  whose  tangent  is 

b  Cos  0+a  Sin  J=    a  +  Tan  6 
a  Cos  0  —  b  Sin  6 


12         PROPAGATION   OF  ELECTRIC   CURRENTS 


tan  i/r+tan  6 
Hence  tan  $  =  1_t^n  .  tan  e  where  tan  \//  =  b/a.    Accordingly 

the  slope  of  (a  -\-  jb)  (Cos  6  +  J  Sin  0)  is  greater  than  the  slope 
of  a  +  jb  by  an  angle  6,  but  the  sizes  are  the  same. 
It  is  proved  in  books  on  trigonometry  that 


and  '      Cos  0=- 


2 

where  e  is  the  base  of  the  Napierian  logarithms  or  the  number 
2-71828  and  j  signifies  \/~^T 

These  are  called  the  exponential  values  of  the  Sine  and 
Cosine,  and  should  be  committed  to  memory.  If  we  substitute 
these  values  in  the  expression  Cos  0  +  j  Sin  0  we  obtain  e  j 0. 
Hence  the  following  are  all  equivalent  expressions  for  a  vector, 
or  complex  quantity,  viz.,  a_ .+  jb,  A  (Cos  0  +  j  Sin  0),  A  e^9  and 
A I  id,  and  they  signify  aline  of  length  A  —  vV  _|_  62  and  slope 

6  =  tan-1  b. 

a 

The  reader  should  practise  himself  in  converting  from  one 
form  to  the  other. 

Ex. — Given  3  +  j  4.     Convert  to  the  other  forms. 

Answer.— The  size  is  A/32  +  42  =  5  =  A    and   0  -  tan~'| 

=  53°  7' 30"  nearly.  Hence  Cos  0  =  0.6,  and  Sin  0  =  0.8.  There- 
fore 5  (0.6  +j  0.8)  and  5  eJ  <53° 7'  30">  or  5/53°  7'  30"  are  equivalent 
to  the  given  expression  3  +  j  4. 

We  have  next  to  consider  the  multiplication  of  two  or  more 
complexes.  If  a  +  jb  ~  A  ej&  is  one  complex  and  ai  +  jbi  — 
AI  €j6i  is  another,  then  the  products  (a  -f  jb)  (ai  +  j^i)  = 
A  AI  €^e  +  ei\  The  rule  then  is,  multiply  the  sizes  of  the  vectors 
and  add  the  slopes.  Thus  the  product  of  a  +  jb  and  ai  -\-  jbi 
is  a  vector  of  which  the  size  is  A/a2  +  &  A/tfi2  +  V  and  the 
slope  is  an  angle  whose  tangent  is  <£  such  that 


a  '  at 
It  follows  that  the  quotient  of  one  complex  quantity  by  another 


MATHEMATICAL   INTBODUCTION  13 

is  obtained  by  the  rule,  divide  the  sizes  and  subtract  the  angles. 
For  if  A  tje  is  one  vector  and  A\  tjei  is  the  other,  then 


Again,  a  complex  is  reciprocated  by  reciprocating  the  size  and 
reversing  the  angle.     For 


Also  we  obtain  any  power  of  a  complex  by  the  rule,  raise  the  size 
to  that  power  and  multiply  the  slope  by  that  power.  Thus  if 
A  tje  is  a  complex  then  its  square  is  A2  €j2e  and  its  square  root 

—     '-  -   j- 

is  VA  €  2  and  nth  power  is  An  e  jn  e  and  nth  root  is  A*  e  ». 

It  will  be  seen,  then,  that  addition  and  subtraction  are  most 
easily  carried  out,  when  the  complexes  are  in  the  typical  form 
a  +  jb,  but  multiplication,  division,  and  raising  to  powers  or 
extracting  roots  when  the  complex  is  in  the  form  A  cj0.  Accord- 
ingly it  is  constantly  necessary  to  convert  from  one  form  to  the 
other  for  calculation. 

If  we  have  any  function  of  complex  quantities  formed  of  the 
products,  powers,  quotients,  or  roots  of  complex  quantities 
such  as 


it  is  not  necessary  to  go  through  the  laborious  process  of  reducing 
it  to  the  canonical  form  A  +  JB  and  to  find  the  size  VA2  +  £2. 
It  follows  at  once  from  the  rules  already  given  that  the  size  of 
the  product  of  two  complexes  is  the  product  of  their  respective 
sizes,  also  that  the  size  of  any  power  of  a  complex  is  the  same 
power  of  its  size,  and  hence  the  size  of  the  quotient  of  two 
complexes  is  the  quotient  of  their  sizes.  It  is  quite  easy  to 
prove  by  actual  multiplication  that  the  size  of  the  vector 
(a  -|_  ji>)  (c  _|_  jrf)  is  \/a2  +  I)1  \/c2  +  d2,  or  is  the  product  of  the 
sizes  of  the  separate  vectors. 


Also  that  the  size  of  ^+L  is  -    "     ^.     Hence  we  can  write 
down  at  once  the  size  of  the  complex  function  (1),  for  it  is 


14    PROPAGATION  OF  ELECTKIC  CUBRENTS 

The  reader  should  work  the  following  exercises  to  familiarise 
himself  with  these  complex  calculations. 

Ex.  1. — Draw  the  two  vectors  3  +  j  4  and  6  4-  j  8  and  give 
their  product  and  quotient  of  the  last  by  the  first  in  the  forms 
(a  +  jb)  and  Va2  +  b*  [0. 

Ans. — The  first   is  a  line  of    length    5  sloping   at   an   angle 

4 
tan-1  £-=  53°  1'  30",  and  the  second  is  a  line  of  length  10  at 

the  same  angle.  Hence  they  are  represented  by  5/53°  1'  30"  and 
10/53°  7'  30".  Their  product  is  a  line  50/106°  15',  and  their 
quotient  is  a  horizontal  line  of  length  2.  Hence  their  product 
is  —  14  +  j  48  and  quotient  2  +  j  0. 

Ex.%. — What  is  the  size  of  the  vector  \J  $  +^  n  ? 

2        3    * 

Ans- 


Ex.  3.— Find  the  square  root  of  the  vector  60  +  j  80  in  the 
form  A/_0. 

Ans.— 10/26°  33'  45". 

Ex.  4. — Show  how  to  calculate  the  value  of  e  the  base  of  the 
Napierian  logarithms. 

Ans. — By  the  exponential  theorem  we  have 

x1  xz 

eg =l-j- #-}_—- —  -f  +  etc 

Hence  if  x  =  1 

1  1 


Hence  e  =  2  +    +   +     +++  etc.  =  2-71828     .     .     . 

The  reader  should  notice  that  each  term  of  the  expansion  of 
€*  is  the  differential  of  the  next  succeeding  term.  Hence  it 

follows  that  ^  (ea-)  =  fx  and  ^ **=**• 

If  we  have  any  vector  or  complex  quantity  represented  in  the 
form  A.tje  or  -A&*  where  pt  is  a  phase  angle  and  t  denotes 
time,  then  the  successive  differential  co-efficients  with  regard  to 
time  are  obtained  by  multiplying  the  function  by  jp,  —  p**,  —  jp3, 


MATHEMATICAL   INTBODUCTION  15 

.-f-j>4,  etc.  Also,  since  the  horizontal  and  vertical  steps  of  the 
vector  are  A  Cos  pt  and  A  Sin  pt,  which  are  simple  periodic 
quantities  as  t  continuously  increases,  it  is  more  convenient  to 
operate  in  mathematical  work  with  the  function  Ae-^  and  to 
take  this  as  the  symbolical  representation  of  a  simple  periodic 
quantity  or  sine  curve  alternating  current,  understanding  this  to 
mean  that  the  periodic  variation  of  the  horizontal  or  vertical 
steps  of  Ac-"**  represents  the  current  at  any  instant. 

We  shall  see  that  it  considerably  simplifies  the  mathematics 
of  alternating  currents  to  deal  only  with  the  maximum  values 
and  avoid  the  cumbersome  trigonometrical  expressions  involved 
if  we  deal  with  the  time  variations  of  the  current  throughout 
the  period.  Hence  in  our  discussions  an  alternating  current  or 
electromotive  force  will  be  represented  by  a  complex  quantity 
such  as  a  -\-jl)  or  Ae^,  and  this  is  to  mean  that  the  vector  or 
line  represented  by  these  complexes  is  to  represent  by  its  length 
the  maximum  value  and  be  supposed  to  revolve  round  one 
extremity  so  that  its  projection  on  a  vertical  line  through  the 
origin  represents  the  actual  value  of  the  periodic  quantity  at 
that  instant  on  the  same  scale  that  the  line  itself  which  revolves 
represents  the  maximum  value  or  amplitude  of  the  alternating 
current  or  E.  M.  F. 

4.  Hyperbolic  Trigonometry. — Since  many  of  the 
mathematical  expressions  involved  in  the  theory  of  the 
flow  of  alternating  currents  through  cables  can  be  most  con- 
veniently presented,  for  the  purposes  of  arithmetic  calculation, 
in  forms  involving  hyperbolic  trigonometry,  it  is  neces- 
sary to  explain  briefly  the  nature  and  properties  of  these 
functions.  Ordinary  trigonometry  is  called  circular  trigonometry 
because  the  mathematical  expressions  employed,  such  as  Sines 
and  Cosines,  are  functions  of  angles  expressed  in  circular 
measure  or  in  their  equivalent  in  degrees.  These  quantities 
may  also  be  regarded  as  functions  of  the  area  of  circular  sectors. 
The  shaded  area  in  Fig.  8  represents  a  segment  of  a  circle. 

The  area  of  this  segment  is  equal  to  ^  r2  0,  where  6  is  the  angle 
PON  in  circular  measure  and  r  is  the  radius  OP.  If  we  call 


16         PROPAGATION   OF   ELECTEIC   CURRENTS 

this  area  u  we  have  2  u/r2  =  0.  Now  Sin  0  =  PM/OP 
and  Cos  0  =  OM/OP. 

Hence  if  we  denote  PM  by  y  and  OM  by  a-, 

„.    2u    y  2u    x 

8m  — r  =  -  and  Cos  -jr=-. 

Accordingly  the  Sine  and  Cosine  are  here  seen  to  be  numerical 
ratios  of  the  sizes  of  two  lines,  and  these  ratios  are  functions  of 
a  certain  kind  of  the  area  and  radius  of  a  circular  sector,  the 


FIG.  8. 

said  lines  being  the  co-ordinates  of  the  upper  point  denning  the 
size  of  the  circular  sector. 

Now  the  hyperbolic  functions  with  which  we  shall  be.  concerned 
are  similar  functions  of  the  area  of  the  hyperbolic  sector  of  an 
equilateral  hyperbola,  and  these  functions  are  related  to  the 
rectangular  hyperbola  in  the  same  manner  that  the  ordinary 
trigonometrical  functions  are  related  to  the  circle. 

We  shall  begin,  therefore,  by  considering  the  mode  of  description 
and  the  equation  of  the  hyperbola. 

The  circle  is  a  curve  described  by  a  point  which  moves  so 
that  its  distance  from  a  fixed  point  called  the  centre  is  constant. 

The  ellipse  is  a  curve  described  by  a  point  which  moves  so 
that  the  sum  of  its  distances  from  two  fixed  points  called  the 
foci  is  constant. 


MATHEMATICAL   INTRODUCTION  17 

The  hyperbola  is  a  curve  described  by  a  point  which  moves 
so  that  the  difference  of  its  distances  from  two  fixed  points  called 
the  foci  is  constant.  Hence  it  may  be  described  mechanically  as 
follows : — On  a  sheet  of  paper  take  two  fixed  points  F,  F'  and 
provide  a  straight  edge  rule  and  a  piece  of  inextensible  thread 
shorter  than  the  rule  by  a  certain  amount. 

Fix  the  rule  so  that  one  end  is  pivoted  on  one  of  the  given 
points  and  fasten  one  end  of  the  thread  to  the  other  fixed  point 


FIG.  9. — Description  of  an  Hyperbola. 

and  attach  the  second  end  of  the  thread  to  the  free  end  of  the 
rule.  Then  press  the  thread  up  against  the  edge  of  the  rule 
with  the  point  P  of  a  pencil  and  revolve  the  rule  radially 
round  one  fixed  point  whilst  keeping  the  thread  pressed  up  to  its 
edge  by  the  pencil  (see  Fig.  9).  The  point  of  the  pencil  will 
describe  one  branch  of  a  hyperbola,  and  the  other  branch  can 
be  described  by  reversing  the  attachments  of  the  thread  and 
rule. 

The  fixed  points  F  and  F'  (see  Fig.  10)  are  called  the  foci  of  the 
hyperbola,  and  the  points  A  A'  where  the  line  FF'  cuts  the  branches 

E.G.  c 


18 


PROPAGATION  OF  ELECTRIC  CURRENTS 


are  called  the  vertices.  The  point  0  bisecting  A  A'  is  called  the 
centre.  The  length  OA'  is  called  the  semi-major  axis  and  is  denoted 
by  a.  The  distance  OF  =  OF'  =  c  is  called  the  focal  distance. 
The  distance  \/c2  —  a2  =  b  is  called  the  semi-minor  axis.  Then 
AF  =  c  -  a  and  AF'  =  c  +  a.  Hence  AF  .  AF!  =  c2-a2=tf. 
If  then  P  is  a  point  on  the  hyperbola  the  difference  of  the 


FIG.  10. — An  Hyperbola. 

distances  PF'  and  PF  is  constant  and  is  equal  to  2a.  Therefore 
PF'—PF—^a,  and  if  x  and  y  are  the  co-ordinates  of  7^  we 
have 

PF=  ^-\-(x  —  cY  and  PF'  =  Jy2+(x+c)'2. 

Therefore  (PF'  +  PF)  (PF'-PF)  =  ±cx     .        .        .     (3) 

and 

Accordingly 
or 


=,  and  PF'-PF=2a, 
a 


MATHEMATICAL   INTRODUCTION 


19 


Substituting  these  last  values  of  PF&ud  PF'  in  the  equation  (4) 
we  have 


or 
or 


.     (5) 


This  last  is  the  equation  to  the  hyperbola  with  origin  at  the 
centre  and  rectangular 
axes  through  the  centre. 
It  is  convenient  to  write 
it  in  the  form 


(7) 


FIG.  11. 


We  have  in  the  next 
place  to  obtain  an  ex- 
pression for  the  area  of 
the  hyperbola  between 
the  vertex  and  any 
ordinate. 

The  expression  for 
the  area  of  an  ele- 
mentary slice  of  the 
hyperbola  contained 
between  two  ordinates 
of  mean  value  y  sepa- 
rated by  a  small  interval 
dx  is  ydx.  Hence  the  area  of  the  hyperbola  between  the  vertex 
and  any  abscissa  x  is  obtained  when  we  know  the  value  of  the 

fx  I)  Cx 

integral      ydx,  or  the  value  of  the  integral  -  U/z2-^  dx. 

Jn'  •'a 

Let  P  be  any  point  on  the  hyperbola  (see  Fig.  11)  and  let  the 
dotted  area  APM  be  denoted  by  A,  then 

A  =  -  T  Vz2-a2  dx    .         .         .  (8) 

a  Ja 

We  have  then  to  find  the  value  of  the  integral  I V  x2  —  a2  dx. 

Now  f  J¥=*dx=  \  ^W  f-y^  (9) 

J  J   *jx*-a2'         J   *Jx2-a2 

c  2 


20         PROPAGATION   OF  ELECTRIC   CURRENTS 


Also 


=x  sltf^tf  -  J  -^p^  ^  -        -  (10) 


This  last  is  obtained  by  noting  that 
d  , 


Hence  adding  (9)  and  (10)  and  dividing  by  2  we  have 


Therefore  we  have 

b  fx        _          xy     ab          \x     y] 

ct>  Ja  ~  2       2        e  (a     b\ 

If  we  draw  the  line  OP  then  the  area  OA P  (shaded)  is  called 
the  hyperbolic  sector  and  is  denoted  by  S. 

It  is  obvious  that  the  area  of  the  triangle  OMP  ( =  3  Xljj  is 
equal  to  the  sum  of  S  and  the  dotted  area  AMP,  which  we  have 

•     7        /»£  ' 

denoted  by  A,  which  last   is  equal  to  -  I   *Jx*  —  a2  dx.     Hence 
we  have 


.        .        .(12) 

If  then  we  consider  a  rectangular  hyperbola  or  one  in  which 
a  =  b  we  have 


2  S 
Finally  denoting  —^  by  it  we  have 

<•-?+*. 


The    ratio  -  is   called    the    hyperbolic    Sine    of   u    and  -  is 


is   called    te      yperoc       ne    o     u    an 

a 

called  the  hyperbolic  Cosine  of  n,  and  these  are  written  Sink  u 
and  Ccsh  u  respectively.     Therefore 

e^Coshtt+Sinhtt    ....    (14) 
Now  the  equation  to  the  hyperbola  is 


MATHEMATICAL   INTKODUCTION 


21 


and  the  equation  to  the  rectangular  hyperbola  is  therefore 

*_2_2/3_i 
- 

or 


Dividing  this  last  equation  by  the  equation  (14)  we  have 


w-Sinh  u 
and  therefore  from  (14)  and  (16)  we  obtain 


Sinh  u  = 


-,  Cosh  u  = 


(15) 


(16) 


(17) 


We  have  therefore  two  definitions  of  Sinh  u  and  Cosh  u  which 
are  consistent  with  each  other. 

Other  hyperbolic  functions  are  denned  as  follows.     The  ratio 

—  =  -  is  called  the  hyperbolic  tangent  and  written   Tanh  u. 
Cosh  u    x 

The  reciprocal  of  the  hyperbolic  Cosine  is  called  the  hyper- 
bolic secant  and  written  Secli  u,  whilst  the  reciprocals  of  the  hyper- 
bolic Sine  and  hyperbolic  tangent  are  called  the  hyperbolic  cosecant 
and  hyperbolic  cotangent  and  written  Cech  u  or  Cosech  u  and 
Coth  u  respectively.  Hence  we  have, 

y  _€»-€- 


Bmh  •«-*-- 

a 

Cosh  u  =  -  =  - 
a 


Cech  u 


Sech 


Gothic- 


(18) 


These  hyperbolic  functions  are  analogous  to  the  correspond- 
ing circular  functions  in  ordinary  trigonometry,  and  form  the 
basis  of  a  hyperbolic  trigonometry  which  has  many  resemblances 
to  it,  but  is  connected  with  the  rectangular  hyperbola  in  place 
of  the  circle. 


22 


PEOPAGATION   OF   ELECTRIC   CURRENTS 


The  numerical  values  of  Sinh  u,  Cosh  u,  Tanh  u,  etc.,  can  be 
calculated  for  various  values  of  u  as  follows  :— 
By  the  exponential  theorem  we  have 


u* 


u 


ri — 9~S£    1 — 9q 

1  .  A  .^9^     _L  .  Zi  .  O 

But  o  (c"  —  e~tt)  =  Sinh  u,  and  hence 


Sinh?t  =  u 


ir 


.  (19) 
.  (20) 

+  etc. 


1. 2. 3  + 1.2. 3. 4. 5^  1.2. 3. 4. 5. 6. 7 

«  "J 

=  w+TS+l^  +  T7+etc (21) 


__ 
Similarly  since  ^  («1t+e-M)  =  Cosh  w  we  have 


Cosh  w  = 


+++  etc. 


(22) 


If  therefore  we  assign  any  numerical  value  to  u  the  corre- 
sponding values  of  Sinh  u  and  Cosh  u  can  be  calculated  with 

any  desired  accuracy. 
Tables  o?  these  hyper- 
bolic functions  have 
been  calculated  and  are 
to  be  found  in  many 
books.  A  Table  of 
Hyperbolic  Sines  and 
Cosines  or  values  of 
Sinh  u  and  Cosh  u  from 
u  =  0  to  u  =  4  has  been 
calculated  by  Mr.  T.  H. 
Blakesley  and  is  pub- 
lished by  Messrs.  Taylor 
and  Francis,  of  Red 
Lion  Court,  Fleet  Street, 
London,  for  the  Physical 
Society  of  London.  A 
very  useful  Table  of  all 
the  Hyperbolic  Functions  has  been  constructed  by  Dr.  A.  E. 
Kennelly,  based  on  Ligouski's  Tables  published  in  Berlin  in 


FIG.  12.— Circular  Sector. 


MATHEMATICAL   INTRODUCTION 


23 


1890,  which  by  kind  permission  is  reproduced  in  the  Appendix 
of  this  hook. 

Similar  Tables  are  given  in  Geipel  and  Kilgour's  Electrical 
Pocket-hook,  and  in  a  collection  of  Mathematical  Tables  arranged 
by  Professor  J.  B.  Dale,  published  by  Messrs.  Arnold  &  Co. 
Also  a  small  but  useful  Table  of  Hyperbolic  Functions  has  been 
published  by  Mr.  P.  Castle,  called  Five-Figure  Logarithms  and 
other  Tables  (Macmillan  &  Co.,  London). 

The  student  should  endeavour  to  obtain  a  clear  idea  of  the 
mathematical  meaning  of  these 
hyperbolic  functions  and  their 
relation  to  the  ordinary  circular 
trigonometrical  functions.  This 
can  be  done  by  comparing  the 
diagrams  in  Fig.  12  and  Fig.  13. 

In  circular  trigonometry  angles 
are  measured  in  radians  or  frac- 
tions or  multiples  of  a  radian.  An 
angle  POM  is  numerically  ex- 
pressed by  the  ratio  of  the  length 
of  the  corresponding  circular  seg- 
ment PA  to  the  radius  OP  of  that 
circle.  Hence  unit  angle  or  1 
radian  is  an  angle  such  that  the 
length  of  the  arc  is  equal  to  the 
radius. 


FIG.  13. — Hyperbolic  Sector. 


The  measure  of  the  angle,  therefore,  is  a  mere  numeric  or  ratio. 

The  circular  functions  Sine,  Cosine,  etc.,  are  also  ratios  of 
lines,  viz.,  the  ratio  of  the  vertical  projection  PM  of  the  radius 
OP  to  the  radius,  or  of  the  horizontal  projection  OM  to  the 
radius  OP.  These  last  ratios  are  considered  to  be  functions  of 
the  angle  POM.  On  the  other  hand  the  area  of  the  circular 

segment   POA   is  equal  to   |   (OP)2  multiplied   by   the   angle 

POA  =  0  in  circular  measure.     Hence  if  we  call  S  this  area  and 
denote  the  radius  OP  by  r,  then  we  have 


24    PKOPAGATION  OF  ELECTRIC  CURRENTS 

If  we  take  the  radius  r  to  be  unity,  then  the  number  which 
denotes  the  angle  6  is  the  same  as  that  which  measures  the 
area  of  the  circular  segment  POP'.  In  other  words,  if  the  angle 
PO'A  is  a  unit  angle  in  circular  measure,  then  the  area  of  the 
circular  sector  POP'  is  a  unit  of  area  in  square  measure. 

The  unit  angle  is  equal  to  57°  17'  45"  nearly.  Hence  if  we 
set  off  a  circular  sector  with  radius  1  cm.  and  double  angle  POP' 
equal  to  114°  35'  30"  the  area  APOP'  will  be  1  square  centi- 
metre. The  circular  trigonometrical  functions  are  therefore  to 
be  regarded  either  as  functions  of  the  ratio  of  the  arc  to  the 
radius  or  of  the  area  of  the  segment  to  the  square  of  the  radius. 

In  the  same  manner  if  we  draw  a  rectangular  hyperbola  and 
take  any  point  P  upon  it  we  can  set  off  a  hyperbolic  segment 
OPAP'  (shaded  area)  analogous  to  the  area  OPAP'  of  the 
circular  segment.  If  the  radius  OA  is  taken  as  unity  and  if  the 
area  of  the  segment  POAf  is  denoted  by  S  and  OA  by  a,  then 

o  o 

-j-  has  been  represented  by  u,  and  by  analogy  we  may  call  u 

the  hyperbolic  angle. 

The  reader  must  carefully  distinguish  between  the  hyperbolic 
measure  of  an  angle  and  the  circular  measure  of  an  angle. 
Thus  the  circular  measure  of  the  angle  POA  (Fig.  13)  may  be 
called  6.  Its  hyperbolic  measure  is  u, 

Now  0  is  such  that  tan  6  =  -==/      ^  x  an(^  V  are  respectively 


PM  and  OM.     But  -=  Sinhw-and  -  Cosh  u  where  a  =  OA. 

Cb  Qt 

II 

Hence  -  =  tanh  u,  and  we  have  tan  6  =  tanh  u. 
oc 

Thus  for  instance  if  the  point  P  is  so  chosen  on  the  rectangular 
hyperbola  of  semi-axis  OA  —  \  that  the  sector  POA  has  an  area  of 
3  square  unit  or  POP'  has  an  area  of  1  unit,  then  u  =  1.  Now 
the  tables  show  that  for  u  =  1  we  have  tanh  u  =  G'76159,  and 
also  that  tan  37°  17'  30"  =  0'76159. 

Hence  the  angle  POA  in  Fig.  13  in  ordinary  degree  measure- 
ment is  37°  17'  30",  and  in  circular  measurement  it  is  0'651,  but 
in  hyperbolic  measurement  it  is  unity. 

The  hyperbolic  functions  are  therefore  ratios  of  lines  which 


MATHEMATICAL   INTRODUCTION  25 

are  functions  of  the  ratio  of  the  area  of  a  hyperbolic  segment  to 
the  square  of  the  radius. 

5.  Formulae  in   Hyperbolic  Trigonometry.— Just 

as  there  are  certain  relations  between  the  circular  functions  of 
ordinary  trigonometry,  so  there  are  similar  formulae  in  hyperbolic 
trigonometry  which  are  of  great  use. 

Fundamental  relations  in  circular  trigonometry  are 

Cos2<9  +  Sin20  =  l (23) 

Sin  (a  +  b)  =  Sin  a  Cos  b  +  Cos  a  Sin  b     .         .         .     (24) 
Cos  (a  +  b)  =  Cos  a  Cos  b -Sin  a  Sin  b     .         .         .     (25) 

From  the  definitions  Sinh  a  =  ~  (€<t  ~~  €~")  and 

Cosh    a  =  g  (*"  +  *  ~ a)  and  similar  definitions  for   Sinh  b  and 

Cosh  b  it  is  easy  to  prove  by  substitution  that 

Cosh2  6-  Sinh2  0  =  1          ....     (26) 
Cosh20  +  Sinh20=:  Cosh  2(9       .         .         .     (27) 

also  that 

Sinh  (a±b)  =  Sinh  a  Cosh  b  ±  Cosh  a  Sinh  b  .         .     (28) 

Cosh  (a±b)  =  Cosh  a  Cosh  b  ±  Sinh  a  Sinh  6  .         .     (29) 

and  hence  that 

,,       7  x      Tanh  a  +  Tanh  b  /om 

Tanh  (a  ±6)  =  ., — Pfj — ^p-^ — _       ....     (30) 
1  ±  Tanh  a  Tanh  6 

These  formulae  are  easily  verified  by  substituting' for 
Sinh  a,  o(ea  — e~"),  and  for  Cosh  a,  o(effl-|-e-ft),  and  the  same  for 

Sinh  b  and  Cosh  b.  It  will  be  seen  that  the  formulae  are  identical 
in  form  with  the  corresponding  ones  in  circular  trigonometry, 
but  that  in  some  cases  algebraic  signs  are  different. 

With  the  aid  therefore  of  a  table  of  hyperbolic  Sines  and 
Cosines  there  is  no  difficulty  in  calculating  out  the  results. 

It  will  be  well  for  the  reader  to  plot  curves  representing  the 
variation  of  the  hyperbolic  functions  as  the  hyperbolic  angle  in- 
creases and  compare  these  with  the  corresponding  curves  for  the 
circular  functions.  The  curves  in  Fig.  14  represent  the  variation 
of  Sinh  u,  Cosh  u,  and  Tanh  u  as  the  angle  u  increases.  These 
curves  therefore  are  non-periodic  and  do  not  repeat  themselves 
like  the  curves  representing  Sin  0,  Cos  0,  Tan  6. 


26    PEOPAGATION  OF  ELECTRIC  CURRENTS 


In  using  hyperbolic  trigonometry  in  connection  with  the 
solution  of  problems  on  the  propagation  of  electric  currents  in 
conductors  we  shall  find  that  we  meet  with  such  expressions  as 


0-5 


1-0 


3-0 


3-5 


4-0 


1-5  2-0  2-5 

Hyperbolic  Angle. 

FIG.  14. — Curves  representing  the  variation  of  the  Hyperbolic  Functions. 

Sinh  (a+jb),  Cosh  (a  -\-jb),  etc.,  where  a  and  b  are  numerical 
quantities  and  j  as  usual  signifies  \/~—  1.  We  have  then  to  con- 
sider the  meaning  of  such  an  expression  as  Cosh  ja  or  Sinh  ja. 


MATHEMATICAL   INTBODUCTION  27 

g/a  _  €  -./«•  €ja  1   €  —  ja 

If  we  remember  that  Sin  a  =  -  and  Cos  a  '  =  - 

4/  « 

fU  _  €  —  U  €U-\~€~U 

and  also  that  Sinh  u  =  —  —  anc^  Cosh  u  =  —  o  —  it  will  be 


__ 

clear  that  Cosh  ja  =  —  ^  —  an^  therefore  that  Cos  a  is  identical 

with  Cosh  ja.  In  other  words  the  Cosine  of  a  circular  angle  is 
identical  with  the  hyperbolic  Cosine  of  a  hyperbolic  angle  ja. 
This  last  expression  ja  is  called  an  imaginary  angle. 

Hence  the  Cosine  of  a  real  angle  is  equivalent  to  the  hyperbolic 
Cosine  of  an  imaginary  angle.  Again  from  the  exponential 
values  of  Sin  a  and  Sinh  a  it  is  evident  that  j  Sin  a  =  Sinh  ja. 
In  a  similar  manner  the  following  formulae  can  be  proved  :— 

Cos  ja  =  Cosh  a.  Cos  a  =  Cosh  ja. 

S'mja=j  Sinh  a.  j  Sin  a  =  Sinhja  .     (31) 

Tan  ja=j  Tanh  a.  j  Tan  a  =  Tanh  ja. 

If  then  we  meet  with  such  an  expression  as  Sinh  (a  +  jb)  we 
can  expand  it  by  the  ordinary  rule  and  eliminate  the  hyperbolic 
functions  of  the  imaginary  angles  by  the  aid  of  the  above 
expressions.  Thus 

Sinh  (a  +jb)  =  Sinh  a  Cosh  jb  +  Cosh  a  Sinh  jb        .     (32) 
or  Sinh  (a  +J&)  =  Sinh  a  Cos  b+j  Cosh  a  Sin  b    .         .     (33) 

In  the  same  way  we  find 

Cosh  (a-\-jb)  =  Cosh  a  Cos  b+j  Sinh  a  Sin  b.         .     (34) 

It  is  evident  then  that  these  equivalents  for  Sinh  (a  -{-jb)  and 
Cosh  (a  -\-jb)  are  vector  or  complex  quantities  of  the  form 
A  -\-jB  because  the  quantities  such  as  Cosh  a  Cos  b  and 
Sinh  a  Sin  b  which  form  the  A  and  B  terms  are  numerical 
quantities. 

Hence  the  hyperbolic  functions  of  complex  angles  such  as 
a  -{-jb  are  vectors,  such  as  Cosh  a  Cos  b  -{-j  Sinh  a  Sin  b. 

The  quantities  a  -\-jb  when  so  used  may  be  called  complex 
hyperbolic  angles  composed  of  a  real  angle  and  an  imaginary 
angle. 

If  we  divide  Sinh  (a  +  jb)  by  Cosh  (a  +  jb)  we  have  Tanh 
a  +  jb,  and  hence 


Tanh  (a+^)  =  ,    .  .    (35) 

1+j  T 


28         PROPAGATION   OF   ELECTRIC  CURRENTS 

If  we  denote  the  size  of  the  vector  Sinh  (a  +  jb)  by  putting 
brackets  round  it  thus  (Sinh  a  -{-jb)  we  have 

(Sinh  a-\-jb)  =  ^/Sinh-  a  Cos'2  b  +  Cosh2  a  Sin'2  b, 
but  Cos2  6  =  1-  Sin2  b  and  Cosh2  a  =  1  +  Sinh2  a. 

Hence  _ 

(Sinh  a+jb)  =  ^Sinh2  a  +  Sin2  b     .  .     (36) 

also 


_ 
(Cosh  a+jb)  =  ^/Cosh^-Sin2  b  .  .     (37) 

Again  the  slope  of  Sinh  (a  +  jb)  is  an  angle  <p  such  that 

Tan  $  =  Coth  a  Tan  6, 
and  of  Cosh  a  -{-jb  is 

Tan  <£  =  Tanh  a  Tan  b. 

Accordingly  if  any  line  or  vector  a  -{-jb  is  given  drawn  on  a 
diagram  we  can  draw  other  lines  or  vectors  on  the  same  diagram 
to  represent  the  quantities  Sinh  (a  +  jb),  Cosh  (a  +  jb), 

Tanh  (a  +jb),  Sech  (a  -{-jb),  Cech  (a  +  jb),  and  Coth  (a  +  jb). 

It  will  be  frequently  necessary  to  consider  how  such  functions 
vary  as  a  or  b  have  different  magnitudes,  that  is  to  say,  as  the 
size  and  slope  of  a  -{-jb  vary. 

For  example,  find  and  draw  the  hyperbolic  functions  of 
1  +  1-5/. 

We  have 

Sinh  (1+y  1-5)  =  Sinh  1  Cos  1'5+j  Cosh  1  Sin  1-5. 

These  numbers  1*5  and  1  are  therefore  angles  in  circular 
and  hyperbolic  measure  respectively.  Since  TI=.  3-1415  =  180° 
the  angle  in  degrees  corresponding  to  1*5  in  circular  measure 

is  180x^  =  89°  7'  44". 

Hence         Sin  1-5  -  '99988  and  Cos  1-5  =  -01525 
also  Sinh  1  =  1-17520  and  Cosh  1  =  1-54308 

Therefore 

Sinh  (1  +j  1-5)  =  1-1752  x  -01525  +j  (1-5431  x  -99988) 
or  Sinh  (1+;  l-5)  =  -018+./  1-543. 

Hence  the  size     (Sinh  (1+y  1-5))  =  1-54  nearly 

and  the  slope  is        Tan-1    .Q18  =89°  nearly. 
Therefore  Sinh  (1  +y  1-5)  =  1  -54  /89°. 


MATHEMATICAL   INTBODUCTION 


29 


In  the  same  manner  we  can,  from  the  formula 

Cos  (a-\-jb)  =  Cosh  a  Cos  b-\-j  Sinh  a  Sin  6, 
find  that  l    Cos  (I  +j  1-5)  =  '0235  +j  1-175 

Hence  the  size  is  1*38  nearly  and  the  slope  90°  or 
Cosh  (1+y  1-5)  =  1-38  /90°. 

Therefore  Tanh  (1+j  1-5  =  1-11 /FT 

Also  Sech  (1  +j  1-5)  =  '072  /W. 

and  Cech  (1+j  1-5  =  -065/89°  and  lastly 

Goth  (l+j  1-5) -'09  /P. 

We  can  therefore  plot  out  these  vectors  as  in  Fig.  15,  where 
the  firm  lines  repre- 
sent the  hyperbolic 
functions  of  0'5  + 
j  0*8,  which  are  more 
widely  separated  than 
those  of  1  +j  1*5.  In 
this  last  case  the  Sinh 
and  Cosh  fall  so  nearly 
on  each  other  that 
they  cannot  be  shown 
as  separated  lines. 

Again,  we  may  take 
any  given  function 
such  as  Sinh  (a  +  jb) 
and  give  various 

ratios  to   -;   that  is, 

CL 

we  may  suppose  the 
vector  a  +  jb  to  be 
turned  round  its  end 
so  that  whilst  retain- 
ing the  same  size  it 
has  various  slopes, 
and  we  may  examine 
the  corresponding 
variation  in  the  hyper- 
bolic functions. 

The  ordinary  logarithms  of  the  hyperbolic  functions,  that  is 


FIG  .15. — Vectors  representing  Hyperbolic 
Functions  of  PI  =  0*5  +/0-8. 


30          PROPAGATION   OF   ELECTRIC   CURRENTS 

logio  (Sinh  u),  logio  (Cosh  «),  and  logio  (Tanh  u),  were  calculated 
by  Dr.  C.  Gudermann  and  published  in  1833  at  Berlin  in  a 
book  entitled  "  Theorie  der  Potenzial  Cyklisch-hyperbolischen 
Functionen."  Unfortunately  he  only  gives  these  logarithms  for 
values  of  u  between  2  and  12.  A  copy  of  the  book  is  in  the 
Graves  Library  of  University  College,  London.  These  tables, 
however,  facilitate  the  calculation  of  the  hyperbolic  functions  of 
complex  angles,  because  they  enable  us  to  calculate  pretty  easily 
the  values  of  Sinh  a  Cosh  b  and  of  Cosh  a  Sinh  b,  etc.,  and 
hence  of  Sinh  (a  +jb),  Cosh  (a  -\-jb\  etc.,  for  values  of  a  and  b 
between  2  and  12. 

We  can  also  obtain  a  graphical  construction  for  the  vectors 
representing  these  hyperbolic  functions  of  complex  angles  in 
the  following  way. 

In  the  case  of  an  ellipse  of  eccentricity  e  and  semi-axes  a  and  b, 
the  distance  from  the  centre  to  either  focus  being  denoted  by/, 
we  have  the  well-known  relations 

62 

— i  =  l  —  62  or  b"  =  cL*  (L—er)  and  ae  =  f. 

a* 

Hence  by  substitution  we  can  put  the  equation  to  the  ellipse 

£3      7/2 

with  origin  at  the  centre,  viz. :  ^+p  =  1  in  the  form 

0    MO     i      ^      H  r&  /OO\ 

c"  X~  +  l^tfl=f 

If  we  take  /  to  be  unity  and  select  such  a  hyperbolic  angle  a 

that  Cosh  a  =  _,  then  Sinh  a  =  v1^2,  and  the  equation  to  the 
e  e 

ellipse  with  origin  at  centre  then  takes  the  form 

-1  .    (39) 


Again  with  regard   to  the  hyperbola  of  eccentricity  ci,  and 

7     O 

semi-axes  ai  and  61  we  have  -  -^=1  —  ei2, 

MI 

or  &i2  =  of  (e?  -  1)  and  /=  ^  e,. 

If  then  the  focal  distance  /=  1,  and  if  we  take  such  a  circular 

angle  /3  that  Cos  /?=-,  we  can  put  the  central  equation  of   the 


2 


x2     11° 
hyperbola,  viz.,  —  3  —    -  =  1»  m  the  form 


MATHEMATICAL   INTBODUCTION 


31 


^•x"1-—^    =1      .  .  (40) 

cS/Tsri^r1    •      •      •      •  <41> 

If  then  we  have  an  ellipse  of  eccentricity  e  =  -  =  „ —    -  and 

a     Oosn  a 

a  conf ocal  hyperbola  of    eccentricity  e\  =  —  =  p — ^  it  is  clear 
that  they  intersect  at  some  point  P  and  that  the  co-ordinates  of 


FIG.  16. 

this  point  x  and   y  are  obtained   by  solving   as   simultaneous 
equations, 

x2  y* 

Cosh^a  +  Sinh2^1     '  '     (42) 

_ £l ..v/2._._i      ....     (43) 

It  is  obvious  'by  inspection,  having  regard  to  the  fact  that 
Cos2  |3  +  Sin2  /3  =  1  and  Cosh2  a  -  Sinh2  a  =  1,  that  the  solu- 
tions of  (42)  and  (43)  are, 

a  Cos  ft  (44) 

a  Sin  /3        .        .        ,         .     (45) 
because  these  satisfy  the  equations  (42)  and  (43). 


32 


PEOPAGATION   OF   ELECTRIC   CURRENTS 


The  radius  vector  OP  of  the  point  of  intersection  of  the 
ellipse  and  hyperbola  is  expressed  as  a  complex  quantity  by 
x+ j?/=Cosh  a  Cos  (3+j  Sinh  a  Sin  /3  =  Cosh  (a+j/3).  Accordingly 
we  can  set  off  a  line  to  represent  Cosh  (a+  JP)  given  a-j-j/3 
as  follows :  Take  a  horizontal  line  and  any  point  0  in  it  (see 
Fig.  16).  Set  off  distances  OF  OF'  on  either  side  of  O  of  unit 


FIG.  17. 

length.  Set  off  distances  OA,  OC  representing  to  the  same 
scale  the  values  of  Cosh  a  and  Cos  (3  as  given  in  the  Tables. 
Draw  a  line  OB  at  right  angles  to  OA  and  take  a  point  B  in  it 
such  that  BF=OA.  Then  describe  an  ellipse  in  the  foci  F  and 
F'  and  semi-axes  OA,  OB.  This  can  be  done  by  making  a  loop 
of  thread  embracing  the  points  F  and  F',  and  of  length  equal  to 
F'F  +  FB  +  BFf  and  moving  a  pencil  point  round  so  as  to 


MATHEMATICAL   INTRODUCTION  33 

keep  the  thread  tight.  Then  describe  an  hyperbola  with  the  same 
foci  and  semi-major  axis  OC  —  Cos  /3.  The  line  OP  represents  to 
scale  Cosh(a+/j8)  because  it  is  x  -\-j  y,  and  these  have  been 
proved  above  to  be  equal  to  one  another.  It  is  well  known  that 
confocal  ellipses  and  hyperbolas  intersect  each  other  at  right 
angles. 

A  very  similar  construction  enables  us  to  draw  a  vector 
representing  Sinh  (a  +J/3),  having  given  a  -\-  j  /3. 

Draw  vertical  and  horizontal  lines  intersecting  at  O  (see 
Fig.  17).  Set  off  distances  OF'  ,  OF  equal  to  unity  on  the 
vertical  line  on  either  side  of  O.  Set  off  a  distance  OA  equal  to 
Cosh  a  to  the  same  scale  and  a  distance  OC  equal  to  Sin  /3,  and 
with  foci  Fr  and  F  describe  an  ellipse  with  semi-major  axis  OA 
and  an  hyperbola  with  semi-major  axis  OC.  These  will  inter- 
sect at  P.  The^i  OP  represents  Sinh  (a  +  j/3).  Let  the 
co-ordinates  of  P  be  x  and  y.  Then  the  equation  to  the  ellipse  is 

#2     ?/2  b2 

-0+"^=1  and  if  e  is  the  eccentricity  -^  =  1  —  e2.      Also  ae  =  1, 
t>2     a1  J  a2 

and  the  equation  to  the  ellipse  is  therefore 


1  1  —  e2 

but  if  a=    =  Cosh  a,  then  —  ^-  =  Sinh2  a  and  the  equation  takes 

6  6"" 

the  form 

Sinh2  a  +  Cosh2  a^1    ' 

In  the  same  way  we  can  prove  that  the  equation  to  the  con- 
focal  hyperbola  is 

2/2  _*!_-, 


or 

°i  —  •*• 

„  O 

or  ggs cos3/3—  •  (4?) 

The  solution  of  the  equations  (46)  and   (47)  as  simultaneous 
equations  gives  us  the  co-ordinates  of  the  point  P  of  intersection. 
It  is  obvious  that  the  solution  is 

x  =  Sinh  a  Cos  ft ) 
y  =  Cosh  a  Sin  (3  \ 

B.C. 


34   PROPAGATION  OF  ELECTRIC  CURRENTS 

Hence 


a  Cos  p+j  Cosh  a  Sin  /3  =  Sinh  (a+j  ft). 

Accordingly  OP  represents  Sinh  (a  +j/3)  on  the  same  scale 
that  OA  =  Cosh  a  and  OC  represents  Sin  /3. 

It  is  clear  that  since  an  ellipse  of  given  foci  is  denned  by  its 
semi-major  axis  and  the  same  for  the  confocal  hyperbola  we 
might  describe  a  number  of  confocal  ellipses  and  hyperbolas  of 
different  eccentricities  and  affix  to  each  a  numerical  value  a 
and  p  where  a  is  such  a  quantity  that  Cosh  a  numerically 
measures  the  semi-major  axis  of  the  ellipse  and  /3  such  a 
quantity  that  Cos  p  represents  the  semi-major  axis  of  the 
hyperbola,  the  focal  distance  OF  for  all  being  unity.  Then  we 
can  obtain  the  value  of  Cosh  (a  +  j/3)  by  looking  out  the  ellipse 
marked  a  and  the  hyperbola  marked  /3  and  joining  the  point  of 
intersection  with  the  centre,  that  vector  would  then  represent 
Cosh  (a  +  j  p).  Such  a  series  of  confocal  ellipses  and  hyperbolas 
has  been  delineated  by  Messrs.  Houston  and  Kennelly  in  a  paper 
entitled  "Resonance  in  Alternating  Current  Lines,"  published  in 
the  Transactions  of  the  American  Institute  of  Electrical  Engineers, 
Vol.  XII.,  April,  1905,  p.  208.  Dr.  Kennelly  has  also  calculated 
the  values  of  Sinh  (a  +j  /3),  Cosh  (a  +  jp),  Tanh  (a  +JP),  Cech 
(a  +JP),  Sech  (a  +jP),  and  Coth  (a  +J/3)  for  fifteen  values  of 

/3 

from  0  to  1-5  and  for  values  of  -  equal  to  1,  2,  3,  4,  10, 


and  set  them  oat  in  Tables1  which  by  his  very  kind  permission 
are  reproduced  here. 

Thus,  for  instance,  the  Table  I.  shows  us  that  the  hyperbolic 
sine  of  a  vector  1/45°  of  which  the  size  therefore  is  unity  and 
ratio  /3/a  is  also  unity  or  slope  45°  is  a  vector  T0055  /54°  32', 
and  from  Table  II.  we  find  that  the  hyperbolic  Cosine  of  the 
same  vector  is  a  vector  1*0803  /27°  29'. 

These  Tables  will  be  found  of  great  use  in  subsequent 
calculations. 

If  then  we  are  given  any  vector  within  limits  in  the  form 
«  +  jb,  we  can  convert  it  into  the  form  Va2  +  fr2  /Tan  -lbja  and 

1  See  Dr.  A.  E.  Kennelly.  "The  Distribution  of  Pressure  and  Current  over 
Alternating  Current  Circuits,"  Tlie  Harvard  Entfnicenncj  Journal,  1905  —  190(5. 


MATHEMATICAL   INTRODUCTION 


35 


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MATHEMATICAL   INTRODUCTION 


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MATHEMATICAL   INTRODUCTION  41 

look  out  in  these  Tables  the  hyperbolic  functions  and  thus 
determine  Sinh  (a  +  jb),  Cosh  (a  +jb),  etc.,  in  the  form  of 
vectors  expressed  as  A  /0,  etc. 

We  sometimes  require  an  expression  for  an  inverse  hyperbolic 
function  such  as  Cosh"1  (a+jb).  Since  this  quantity  is  a  vector 
it  must  have  such  a  value  that 

Cosh"1  (a+jb)=x+jy, 
or  Cosh  (x  +jy)  =  a+jb. 

Hence  a+jb  =  Cosh  x  Cos  y+j  Sinh  x  Sin  y. 

Equating  vertical  and  horizontal  steps  we  have 
a  =  Cosh  x  Cos  y 
b  —  Sinh  x  Sin  y. 

But  Sin2  y+  Cos2  y  =  l  and  Cosh3  x-  Sinh2  x  =  l. 

Therefore  by  substitution  we  find 

-^2-+— ^-  =  1 
Cosh   x     Sinh  x 

a2  62 

or  — 2~+~  — 2 —  — =1. 

Cosh  x     Cosh  x  —  1 

Multiplying  up  we  arrive  at  a  biquadratic  equation 

Cosh4  x  —  Cosh  x  +  a2  =  a3  Cosh  x  +  b*  Cosh  x 
which  can  be  written  in  the  form, 


a 

{Cosh  *  ---  j— 


Hence 


This  last  expression  can  be  put  in  the  form 


which  is  an  exact  square.     Therefore 


Coshz  =  -  ^p  .        .     (49) 

In  the  same  manner  we  can  show  that 

Cos  y  —  ^*         '    ' — — 
2 


42          PROPAGATION   OF   ELECTRIC   CURRENTS 

Accordingly 

Cosh      (a  -f  jb)  = 


.      .    (SO) 


And  by  a  similar  process  we  can  prove  that 


.4 

These  formulae  have  important  applications. 


CHAPTER  II 

THE  PROPAGATION  OF  ELECTROMAGNETIC  WAVES  ALONG  WIRES 

1.  Wave  Motion.— As  the  subject-matter  of  these  lectures 
is  an  exposition  of  the  effects  connected  with  the  propagation  of 
electromagnetic  waves  along  wires,  it  may  be  well  to  commence 
by  some  explanation  of  the  nature  of  wave  motion  generally. 
Let  us  consider  a  material  medium  like  the  air  composed  of  dis- 
crete particles  or  atoms  which  we  shall,  for  the  sake  of  simplicity, 
assume  to  be  initially  at  rest.  The  medium  has  two  fundamental 
mechanical  qualities.  It  possesses  Inertia  in  virtue  of  which  any 
particle  of  it  when  set  in  motion  tends  to  persist  in  that  motion 
unless  compelled  to  change  its  motion  by  impressed  force.  This 
is  equivalent  to  stating  that  when  any  mass  M  of  the  medium  is 
moving  without  rotation  with  a  velocity  V  it  possesses  kinetic  or 

motional  energy  measured  by  ^  MV2.    Also  the  medium  possesses 

some  kind  of  Elasticity — that  is  it  resists  change  of  form  or 
shape  or  motion.  In  the  case  of  a  fluid  like  air  the  elasticity  is 
resistance  to  change  of  volume  of  a  given  mass.  It  resists  com- 
pression or  expansion.  In  consequence  of  these  two  qualities 
inertia  and  elasticity  the  medium  permits  the  propagation 
through  it  of  ware  motion.  This  means  that  any  change  in  the 
medium  made  suddenly  at  one  place  is  not  instantly  reproduced 
or  repeated  at  all  points,  but  makes  its  appearance  successively 
at  different  points.  Thus,  if  in  an  unlimited  mass  of  air  we  cause 
a  sudden  increase  in  pressure  of  the  air  at  one  spot  by  heating  it, 
say  by  an  electric  spark,  the  surrounding  air  does  not  imme- 
diately relieve  this  pressure  by  moving  outwards  everywhere  at 
once,  because  in  virtue  of  the  inertia  of  the  air  the  force  due  to 
the  initial  compression  cannot  immediately  create  outward 
motion  in  the  surrounding  shell  of  air.  When,  however,  the 


44 


PEOPAGATION  OF  ELECTKIC  CURRENTS 


immediately  surrounding  layer  of  air  has  been  set  in  motion 
outwards  it  relieves  the  pressure  at  the  origin,  and  the  original 
state  of  compression  is  now  transferred  to  a  shell  of  air 
embracing  the  original  region  of  compression.  This  process  again 
repeats  itself,  and  the  state  of  compression  is  handed  on  to  a  still 
larger  spherical  shell  or  layer,  and  thus  the  original  state  of  com- 
pression is  propagated  outwards  in  the  form  of  a  spherical  shell 
of  compression  which  changes  its  locus  progressively  by  con- 
tinually increasing  its  size. 

Whilst  the  general  body  of  the  air  remains  undisturbed  this 
thin  spherical  region  or  shell  in  which  the  air  is  compressed 


CL 


CL' 


,r  -- 


y 


PIG.  1. 

continually  becomes  greater  in  radius  and  forms  what  is  called  a 
wave  of  compression  in  the  air. 

The  characteristic  of  wave  motion  is  therefore  that  the 
particular  kind  of  disturbance  (in  this  case  compression)  is 
repeated  successively  and  not  simultaneously  at  all  points  of 
the  medium.  If  we  take  two  points  in  the  medium  separated 
by  a  certain  distance  x  and  note  the  time  interval  t  between 
the  appearance  of  the  disturbance  at  these  places,  then  x/t  is 
called  the  icave  velocity  (W).  This  wave  velocity  depends  upon 
the  specific  qualities  of  the  medium,  viz.,  its  density  or  inertia 
per  unit  of  volume  and  its  elasticity. 

To  fix  our  ideas  let  us  consider  waves  of  longitudinal  dis- 
placement such  as  sound  waves  travelling  up  a  tube  of  unit  cross- 
section  filled  with  air.  The  particles  of  air  lying  on  any  section 
of  the  tube  will  then  move  to  and  fro  together.  Let  the  density 


ELECTROMAGNETIC   WAVES   ALONG  WIRES        45 

or  mass  of  air  per  unit  volume  be  denoted  by  />,  and  its  elasticity 
or  the  ratio  of  compressing  force  or  pressure  to  the  corresponding 
compression  in  volume  be  denoted  by  e.  Then  if  dp  is  the 
increment  of  pressure  causing  a  reduction  of  volume  dr  in  a 

volume  of  air  r,  we  have  c  =  -  ^  .      Consider  a  layer  of  air 

particles  lying  on  a  section  a  b  of  the  air  in  the  tube  (see  Fig.  1). 
Let  x  denote  their  distance  from  a  fixed  section  at  zero  time,  and 
let  x  +  y  be  their  distance  after  a  time  t  as  the  wave  of  longi- 
tudinal displacement  moves  over  them.  Then  y  is  the  displace- 
ment in  the  time  t  of  the  particles  which  form  this  section  ab. 

Suppose  then  that  we  fix  our  attention  upon  a  slice  of  the  air 
bounded  by  two  planes  at  distances  x  and  x  -f-  bx  from  the  origin. 
As  the  wave  passes  over  this  slice  the  sections  of  it  are  moved 
so  that  the  particles  which  were  initially  at  x  are  moved  to 
x  +  y,  and  the  particles  which  were  initially  at  x  +  dx  are 
moved  to 

x+y  +  S 


Hence  the  thickness  of  the  slice  which  was  originally  bx  becomes 
bx  +  by.     Its  increase  in  volume  is  therefore  by,  and  the  ratio 

of   increase  of   volume  to  original  volume  is  ^-,  or  ultimately, 

when  bx  is  very  small,  it  becomes  ^|. 

If  the  changes  in  pressure  of  the  slice  of  air  are  made  very 
slowly,  then  the  product  of  pressure  p  and  volume  r  of  a  unit  of 
mass  is  constant,  which  may  be  expressed  by  the  formula 
pr  —  a  constant.  If,  however,  the  compression  is  very  suddenly 
applied  so  that  the  heat  due  to  the  compression  remains  in  the 
slice  and  augments  its  pressure  or  elasticity,  then  the  relation  of 
p  and  v  is  given  by  the  equation  pva  =  a  constant  where  a  =  T41 
nearly,  and  is  the  ratio  of  the  specific  heat  at  constant  pressure  to 
the  specific  heat  at  constant  volume.  This  is  the  case  in  an  air 
wave.  Hence  we  have  by  differentiation  of  pva  =  constant, 

dpta  +  ai'a~l  pdt  =  0  or  dp  —  —  ap  —  or  —    -,  •  =  e  =  —  ap. 

The  force  moving  the  slice  of  air  of  thickness  bx  is  the 
difference  of  pressure  on  its  two  surfaces,  viz.,  the  value  of 


46          PEOPAGATION   OF   ELECTRIC   CURRENTS 

-=-(dp)8x.      But<fy>=  —  ap —  and  we  have  shown  that  for  the  air 

dx"  '  L  v  ' 

motion  here  considered  we  have  — ~^m 

Hence  the  moving  force  on  the  air  section  is 


The  displacement  of  the  slice  being  y,  it  follows  that  its 
acceleration  is  ~/T  and  since  its  mass  is  pbx,  the  equation  of 
motion  is 


.....   W 

The  above  is  a  type  of  differential  equation  which  presents  itself 
very  frequently  in  Physics.  It  is  not  difficult  to  show  that  it  is 
satisfied  by  any  value  of  y  which  is  made  up  of  the  sum  of  any 

single  valued  functions  of  x  —  \J  -  t  and  x  +  \/-  t. 

*  P  *  P 

So  that  y=F  (x~\/e-  *)  +  F  (x  +  \/e  *)         .        •     (2) 

Any  function   such   as   F  (x-^/e  t\  represents  a  wave  of 

wave-form  y  =  F  (x)  travelling  forward  with  a  velocity  JF=/y/-. 

For  the  function  F  (x  —  \/  e-  M  has  the  same  value  if  for  x  we 

\       v       / 


substitute  x  —  x',  and  for  f,  t  —  t',  provided  that  x'/t'  =*\~>    The 

reader  should  carefully  consider  the  physical  meaning  of   this 
statement. 

Any  function  of  x  such  as  y  =  F  (x)  represents  a  stationary 
curve  whose  ordinate  y  at  any  point  is  some  function  of  its 
abscissa  x.  It  therefore  represents  a  wave-form.  If  the  curve 
moves  bodily  forward  without  change  of  shape  with  a  speed  W, 
then  the  ordinate  having  a  value  y  at  a  time  t  corresponding  to 
an  abscissa  x  has  the  same  height  as  the  ordinate  y  at  a  distance 


ELECTROMAGNETIC    WAVES   ALONG   WIRES        47 

f]  'J* 

x  -f  tfo  corresponding  to  a  time  t  +  dt,  provided  -j-  is  the  velocity 

ctt       . 

W  of  the  wave. 

In  other  words,  the  characteristic  of  a  wave  motion  is  that  the 
same  state  is  repeated  at  a  distance  dx  ahead  at  a  time  dt  ahead 

provided  v.-  is  the  velocity  of  disturbance.  Hence  any  mathe- 
matical function  such  as  F  (x  —  JJY)  for  which  this  is  true 
represents  a  wave  advancing  with  a  velocity  jr. 

Accordingly  for  a  medium   of  density  p  and  elasticity  e  the 
wave  velocity  W  is 

; — 

e 


V?   ' 

2.  The  Electromagnetic  Medium.— It  is  now  almost 
universally  agreed  that  the  phenomena  of  electricity  and 
magnetism  render  it  necessary  to  postulate  an  electromagnetic 
space-filling  medium  or  aether,  and  it  has  been  shown  that  what 
we  call  light  and  radiant  heat  as  well  as  electric  radiation  are 
waves  of  a  particular  kind  in  this  medium.  Moreover,  a  large 
body  of  proof  exists  tending  to  show  that  the  elements  of  material 
substance  described  as  atoms  are  built  up  of  constituents  called 
negative  electrons  or  corpuscles  and  of  positive  electrons ;  and 
that  these  negative  electrons  collectively  constitute  so-called 
negative  electricity.  The  reader  desirous  of  placing  himself 
an  courant  with  what  is  known  and  believed  on  these  matters 
may  be  referred  to  the  following  excellent  works  for  a  full 
exposition  of  them : — "  Electricity  and  Matter,"  by  Sir  J.  J. 
Thomson  (Archibald  Constable  &  Co.,  London);  "  A  Treatise 
on  Electrical  Theory,"  by  G.  W.  de  Timzelmann  (Charles 
Griffin  &Co.);  "The  Electron  Theory,"  by  E.  E.  Fournier 
d'Albe  (Longmans,  Green  &  Co.)  ;  "  Electromagnetic  Theory,"  by 
Oliver  Heaviside  (The  Electrician  Publishing  Company, London). 
The  advanced  reader  will  do  well  to  consult  "^Sther  and 
Matter,"  by  Sir  Joseph  Larmor  (Cambridge  University  Press), 
and  "  The  Theory  of  Electrons,"  by  H.  A.  Lorentz  (David  Nutt 
&  Co.,  London). 

The  sum  and  substance  of  the  scientific  creed  taught  by  these 
writers  is  that  the  basis  for  all  physical  phenomena  as  well  as 


48    PROPAGATION  OF  ELECTKIC  CURRENTS 

the  source  of  all  gravitative  Matter  is  to  be  found  in  the  pro- 
perties of  the  Universal  ^Ether,  and  that  not  only  Matter  but  also 
Electricity  has  an  atomic  structure,  and  that  the  atoms  of 
electricity,  or,  to  speak  more  correctly,  of  negative  electricity, 
are  the  electrons  which  are  the  constituents  of  the  chemical 
atom. 

The  hypothesis  has  been  advanced  that  the  electron  itself  is 
a  strain  centre  or  focus  of  certain  lines  of  strain  in  the  aether  of 
a  particular  kind.  Hence  the  movement  of  the  electron  is 
merely  a  displacement  of  the  strain  form  from  one  place  to 
another  in  a  stagnant  aether.  Experimentally  it  is  established 
that  an  electron  is  a  small  charge  of  negative  electricity  assumed 
to  be  distributed  over  a  small  sphere  having  a  diameter  about 
one  hundred  thousandth  of  that  of  a  chemical  atom.  It  is 
therefore  a  centre  on  which  converge  lines  of  electric  force.  The 
phenomena  of  electricity  and  magnetism  prove  that  in  the 
neighbourhood  of  electrified  bodies  there  is  a  distribution  along 
curved  or  straight  lines  of  electric  strain,  which  strain  is  a 
physical  state  of  the  material  dielectric  or  the  interpenetrating 
aether.  This  state  is  also  called  electric  displacement  or  polarisa- . 
tion.  Similarly  near  magnetic  poles  and  conductors  carrying 
electric  currents  there  is  a  distribution  of  magnetic  flux  or 
induction. 

The  magnetic  flux  and  electric  strain  are  particular  states  of 
the  aether  or  matter  occupying  the  field,  which  are  vector 
quantities  having  direction  as  well  as  magnitude  at  each  point  in 
the  field.  Thus  the  electron  is  a  centre  of  converging  lines  of 
electric  strain,  and  a  wire  conveying  an  electric  current  is 
embraced  by  endless  lines  of  magnetic  flux.  The  important 
question  then  arises  whether  these  "lines  of  force"  are  merely 
mathematical  abstractions  like  lines  of  latitude  and  longitude  or 
whether  we  are  to  regard  them  as  having  objective  existence. 
Arguments  of  a  weighty  character  have  been  advanced  by  Sir 
J.  J.  Thomson  for  the  view  that  these  lines  of  magnetic  and 
electric  force  are  not  merely  directions  in  the  field,  but,  so  to 
speak,  structures  which  compose  it.1  In  other  words,  not  only 
matter  and  electricity  but  also  electric  and  magnetic  fields  are 

.    i  See  Sir  J.  J.  Thomson,  Phil.  Mag.,  Ser.  6,  Vol.  XIX.,  p.  301,  February,  1910. 


ELECTROMAGNETIC  WAVES   ALONG   WIRES        49 

atomic  in  nature.  Accordingly  the  electron,  as  the  atom  of 
electricity,  is  to  be  thought  of  as  a  centre  on  which  converge  a 
certain  definite  number  of  lines  of  electric  strain,  and  these  lines 
are  in  themselves  states  of  strain  in  the  aether,  analogous  in  some 
sense  to  vortex  filaments  in  a  liquid.  To  employ  a  somewhat 
crude  simile,  the  electron  must  be  thought  of  as  a  ball  from 
which  proceed  in  every  direction  long  hairs  or  filaments  radially 
arranged  which  it  carries  about  with  it. 

Sir  Joseph  Larmor  has  based  an  elaborate  and  consistent 
theory  of  electrical  phenomena  on  the  supposition  that  these 
lines  of  electric  strain  radiating  from  the  electron  as  a  centre  are 
lines  of  torsional  strain  in  the  aether.  He  assumes  the  aether  to 
be  a  continuous  or  extremely  fine  grained  medium,  every  particle 
of  which  resists  absolute  rotation.  This  resistance  to  rotation 
may  proceed  from  a  whirling  motion  of  these  very  small  parts 
which  bestows  a  gyroscopic  stiffness  upon  the  particles.  This, 
however,  is  not  the  place  to  enter  upon  a  discussion  of  aether 
theories ;  the  reader  may  be  referred  to  Sir  J.  Larmor's  book 
11  ^Ether  and  Matter  "  for  a  description  of  a  working  model  of 
this  rotational  aether  based  on  the  well-known  properties  of  the 
gyroscope.  All  we  shall  attempt  here  is  to  provide  such  clear 
conceptions  of  the  working  processes  of  an  electromagnetic  field 
as  shall  assist  the  end  we  have  in  view. 

3.  Electric  and  Magnetic  Forces  and  Fluxes. 

The  region  near  electrified  bodies,  called  an  electric  field,  is 
then  the  seat  of  a  particular  state  called  electric  strain  which  we 
shall  consider  is  located  along  certain  definite  lines  called  lines 
of  electric  strain  or  sometimes  lines  of  electric  force. 

Strictly  speaking  the  electric  strain  is  the  state  in  the  dielectric 
caused  by  an  agency  called  electric  force.  In  the  same  way  the 
region  near  magnets  or  electric  currents,  called  a  magnetic  field, 
is  the  seat  of  magnetic  flux  located  along  certain  lines  called 
lines  of  magnetic  flux.  Electrified  bodies  and  magnetic  poles  or 
electric  currents  exercise  attractive  or  repulsive  forces  on  one 
another  which  can  be  measured  in  absolute  units  or  dynes.  The 
dyne  is  defined  to  be  the  force  which,  after  acting  on  a  mass  of 
1  gram  for  1  second,  gives  it  a  velocity  of  1  centimetre  per 

B.C.  E 


50         PKOPAGATION   OF  ELECTRIC   CURRENTS 

second  in  the  direction  in  which  it  acts.  A  unit  magnetic  pole 
is  one  which  acts  on  another  unit  magnetic  pole  at  a  distance  of 
1  centimetre  with  a  force  of  1  dyne.  If  a  unit  magnetic  pole 
is  placed  in  a  magnetic  field  the  strength  of  the  field  or  the 
magnetic  force  at  that  point  is  measured  by  the  force  in  dynes 
acting  on  the  unit  magnetic  pole  placed  there.  We  shall  denote  the 
magnetic  force  at  any  point  in  a  field  so  measured  by  the  letter 
H.  The  direction  of  the  lines  of  magnetic  flux  in  a  field  can  be 
mapped  out  by  means  of  iron  filings.  In  the  case  of  a  wire 
carrying  a  current  the  lines  of  magnetic  flux  are  closed  lines 
embracing  the  wire.  The  creation  of  an  electric  current  in  a 
conducting  circuit  necessitates  the  existence  in  it  of  some 
source  of  electromotive  force.  If  the  conducting  circuit  is 
interrupted  anywhere,  the  source  of  electromotive  force  still 
existing  in  it,  a  difference  of  potential  is  created  between  parts  of 
it,  and  in  the  non-conducting  region  an  electric  force  is  pro- 
duced tending  to  generate  electric  strain.  The  presence  of  an 
electric  field  is  detected  by  the  existence  of  a  mechanical  force 
acting  on  a  small  positively  electrified  body  placed  in  the  field. 
Two  small  spheres  charged  with  electricity  exert  a  mechanical 
force  on  each  other  which  may  be  measured  in  dynes.  A  unit 
charge  is  one  which  acts  on  another  unit  charge  at  a  distance  of 
1  centimetre  with  a  force  of  1  dyne.  From  a  mathematical 
point  of  view  these  electric  attractions  and  repulsions  can  be 
regarded  as  simply  the  action  at  a  distance  of  electrons — 
negative  electrons  repelling  negative  and  attracting  positive  and 
positive  repelling  positive  and  attracting  negative  ones.  But  as 
an  explanation  of  what  really  happens  modern  scientists  do  not 
admit  action  at  a  distance,  but  only  the  immediate  action  of 
contiguous  parts  of  the  same  medium.  Accordingly  the  forces 
between  electrified  bodies  must  be  sought  for  not  in  actions 
at  a  distance  between  electrons,  but  in  the  immediate  actions 
of  their  associated  lines  of  electric  strain  in  the  universal 
aether. 

It  is  found  that  a  consistent  theory  can  be  built  up  on  the 
assumption  -that  the  lines  of  electric  strain  exert  a  tension  like 
elastic  threads  and  always  tend  to  make  themselves  as  short  as 
possible.  Also  they  exert  a  lateral  pressure,  and  their  arrange- 


ELECTROMAGNETIC   WAVES   ALONG   WIRES        51 

ment  in  a  field  is  due  to  the  conflict  between  their  longitudinal 
tension  and  lateral  pressure. 

An  explanation  of  the  properties  of  lines  of  electric  strain  is 
only  possible  on  the  basis  of  some  theory  of  the  aether,  but  it 
is  possible  to  explain  it  if  we  assume  a  medium  possessing 
inertia  and  some  sort  of  fine  grained  whirling  structure. 

Thus  suppose  a  number  of  thin  inextensible  but  flexible 
spherical  envelopes  or  bags  to  be  filled  with  liquid.  IE  the 


FIG.  2. 

liquid  in  these  bags  is  at  rest  it  will  assume  a  spherical  form,  but 
if  set  in  rapid  rotation  round  an  axis  each  spherical  ball  will 
become  converted  into  an  oblate  spheroid  like  an  orange, 
flattened  at  the  poles  and  expanded  at  the  equator.  If  the  balls 
are  compelled  to  remain  in  contact  with  each  other  and  if  the 
axes  of  rotation  are  arranged  in  parallel  lines,  this  flattening  and 
expansion  of  the  cells  will  cause  the  row  of  spheres  to  become 
shorter  along  the  axis  of  rotation  and  also  by  their  equatorial 
expansion  to  exert  a  pressure  at  right  angles,  as  illustrated  in 
the  diagrams  in  Fig.  2,  in  which  the  circles  represent  the 

E  2 


52          PKOPAGATION   OF   ELECTRIC   CUREENTS 

spherical  bags  which  by  rotation  have  become  spheroids,  thus 
contracting  in  length  along  the  line  of  rotation  and  expanding 
laterally. 

By  some  such  explanation  the  student  will  be  able  to  see  that 
electric  attractions  and  repulsions  can  be  explained  by  these 
properties  of  lines  of  electric  strain.  We  have  to  assume  that 
a  line  of  electric  strain  always  starts  from  a  negative  electron 
and  ends  on  a  positive  one,  unless  it  happens  to  be  self -closed 
or  endless.  Furthermore  we  must  assume  that  in  conductors 
the  electrons  are  quite  free  to  move  or  that  the  ends  of  lines  of 
electric  strain  can  slide  along  the  surface  of  conductors  but 
cannot  so  move  over  the  surface  of  insulators. 

We  have  in  the  next  place  to  consider  the  nature  of  lines  of 
magnetic  flux. 

Addressing  ourselves  first  to  the  facts  we  find  that  a  moving 
charge  of  electricity  or  say  an  electron  creates  a  magnetic  field 
along  circular  lines  whose  planes  are  perpendicular  to  its  line  of 
motion  and  centres  are  on  that  line.  Hence  if  a  spherical 
charge  with  radial  lines  of  electric  strain  moves  forward  it  creates 
circular  lines  of  magnetic  flux  embracing  its  line  of  motion. 
The  magnetic  lines  of  flux  are  perpendicular  to  the  directions  of 
the  lines  of  strain  and  line  of  motion. 

This  was  first  shown  experimentally  to  be  the  case  by 
H.  A.  Rowland  in  1876  and  was  confirmed  by  Rowland  and 
Hutchinson  in  1889  and  also  by  Rontgen  in  1885.  Doubt  was 
thrown  on  the  facts  by  M.  V.  Cremieu  in  1900,  but  Rowland's 
conclusions  were  reaffirmed  by  H.  Fender  in  1901  after  a  careful 
research.1  A  brief  general  description  of  this  classical  experi- 
ment is  as  follows  :— 

A  pair  of  circular  glass  plates  are  covered  with  gold  leaf  which 
is  divided  by  radial  cuts.  These  plates  are  charged  to  a  high 
potential  with  electricity  and  set  in  rapid  rotation  round  their 
centres.  The  two  plates  are  placed  parallel  and  near  to  each 
other.  Between  them  is  suspended  a  sensitive  shielded  magnetic 

1  See  H.  A.  Kowland,  Pogg.  Ann.,  1876,  Vol.  CLVIIL,  p.  487  ;  Rowland  and 
Hutchinson,  Phil.  Mag.,  1889,  Vol.  XXV1L,  p.  445  ;  Rontgen,  Her.  <lcr  liei-Un.  Al«td.. 
1885,  p.  195;  Cremieu,  Comptes  Renting  1900,  Vol.  CXXX.,  p.  1544;  1901, 
Vol.  CXXXL,  pp.  578,  797  ;  Vol.  CXXXII.,  pp.  327,  1108  ;  H.  Fender,  Phil.  Mag., 
1901,  Vol.  II.,  p.  179. 


ELECTROMAGNETIC   WAVES   ALONG  WIRES       5B 

needle.  When  the  charged  plates  revolve  at  a  high  speed  the 
needle  is  deflected  in  the  same  manner  as  it  would  be  if  an 
electric  current  were  flowing  round  the  periphery  of  the  disk. 
If  the  plates  are  charged  positively  the  convection  current,  as 
it  is  called,  has  the  same  magnetic  effect  as  a  voltaic  current 
flowing  round  the  disk  in  the  direction  of  rotation  and  if  charged 
negatively,  in  the  opposite  direction. 

Hence  we  have  an  experimental  proof  that  a  moving  charge  of 
electricity  produces  a  magnetic  field. 

It  follows  that  lines  of  electric  strain  moving  transversely  to 
their  own  direction  create  lines  of  magnetic  flux. 

A  very  beautiful  direct  proof  of  the  fact  that  a  moving  charged 
body  is  equivalent  to  an  electric  current  has  been  given  by  Professor 
R.  W.  Wood.1  When  carbonic  dioxide  gas  strongly  compressed 
in  a  steel  bottle  is  allowed  to  escape  from  a  nozzle  the  sudden 
expansion  creates  a  fall  of  temperature  sufficient  to  solidify  some 
of  the  gas  into  small  particles.  These  particles  of  C02  are 
electrified  by  friction  against  the  nozzle  like  the  particles  of 
water  when  the  steam  escapes  in  Lord  Armstrong's  hydro- 
electric machine.  The  particles  of  solid  carbonic  dioxide  are 
electrified  positively.  If  this  jet  is  sent  along  a  glass  tube  it  is 
possible  to  obtain  velocities  of  the  electrified  particles  as  high  as 
2,000  feet  per  second.  Professor  Wood  found  that  a  magnetic 
needle  suspended  outside  the  tube  was  affected  just  as  if  the  tube 
had  been  a  wire  conveying  an  electric  current. 

In  order  that  we  may  define  more  accurately  the  relation  of 
lines  of  electric  strain  and  magnetic  flux  we  must  attend  to  the 
following  definitions. 

Electric  strain  may  be  said  to  be  produced  in  a  dielectric  by 
electric  force  or  stress  just  in  the  same  manner  that  mechanical 
strain  is  produced  by  mechanical  force  or  stress.  We  call  the 
ratio  of  the  stress  to  the  homologous  strain  the  elasticity  of  the 
material,  and  similarly  we  may  call  the  ratio  of  the  electric  stress 
or  force  (E)  to  the  electric  strain  (D)  the  electric  elasticity. 

Unfortunately  the  term  dielectric  constant  (K)  or  specific 
inductive  capacity  was  the  name  given  a  long  time  ago  to  the 

1  See  PlilL  May.,  1902,  6th  Ser.,  Vol.  II.,  p.  659. 


54         PROPAGATION   OF   ELECTRIC   CURRENTS 

4?r-D 

ratio  -^-.     In  other  words  the  relation  between  the  total  dis- 

±!j 

placement  through  the  surface  of  a  sphere  of  unit  radius  at  the 
centre  of  which  is  placed  a  unit  charge^to  the  electric  force  at  a 
unit  of  distance  has  been  called  the  dielectric  constant. 
Suppose  that  a  quantity  of  electricity  Q  reckoned  in  electrostatic 
units  is  placed  on  an  extremely  small  sphere  and  that  we  describe 
round  its  centre  a  larger  sphere  of  radius  r.  Then  the  surface 
of  this  last  sphere  is  47rr2  and  the  displacement  per  unit  of  area 
or  number  of  lines  of  electric  strain  passing  through  this  sphere 
being  called  D,  the  total  displacement  is  47rr2D,  and  this  is  denned 

to  be  equal  to  Q.     Hence  the  displacement  D=    —2. 

The  electric  force  E  at  a  distance  r  is  ^  where  K  is  the  so- 
called  dielectric  constant.  Hence  the  ratio  of  stress  to  strain  is 
the  ratio  --2  :  T^2=—  ^ne  electric  elasticity  and  the  ratio 


47rD  /  E  =  K  =  the  dielectric  constant.  We  do  not  know  the 
actual  number  of  lines  of  electric  strain  proceeding  from  an 
electron  or  natural  unit  of  electricity,  but  it  is  convenient  to 
consider  that  the  total  number  of  lines  of  electric  strain  pro- 
ceeding from  a  charged  body  is  numerically  equal  to  the  charge. 
Thus  if  the  charge  is  Q  there  are  Q  lines  of  strain  passing  through 
the  surface  of  a  sphere  of  radius  r  described  round  it.  Hence 
the  lines  of  strain  per  unit  area  or  the  density  of  the  lines,  also 
called  the  displacement  D,  is  such  that  4  -n  r2  D  =  Q. 

We  have  next  to  consider  the  relation  between  the  magnetic 
flux  and  the  electric  strain.  The  magnetic  flux  (B)  is  considered 
to  be  an  effect  due  to  magnetic  force  (H),  and  the  ratio  of  the 
flux  to  the  force  is  called  the  magnetic  permeability  (/x).  Hence 
B  =  n  II.  The  magnetic  flux  density  B  signifies  the  number  of 
lines  of  magnetic  force  which  pass  normally  through  unit  of 
area.  Accordingly  we  have  the  two  fundamental  equations  of 
electromagnetism  as  follows  :— 

B=^H       .        .  .     (4) 

D  =  ~E    .....     (5) 


7T 


ELECTROMAGNETIC  WAVES  ALONG  WIRES       55 

The  occurrence  of  this  4  it  in  the  second  equation  is  due  to  the 
mode  of  definition  adopted  for  the  displacement  Z).  It  would 
have  been  preferable  if  the  electric  force  E  had  been  so  defined 
that  the  force  at  a  distance  r  from  a  quantity  Q  were  taken  as 

9  vt  and  then  this  would  have  given  D  =  KE.      Taking, 

4  7T  T"  J\. 

however,  the  usual  definition  we  have  the  relation  as  given  in  the 
equations  above. 

We  have  next  to  consider  the  relation  between  magnetic  flux  B 
and  electric  strain  or  displacement  D.  This  is  based  upon  the 
two  following  facts  : — 

(i.)  That  lines  of  electric  strain  when  moved  laterally  through 
a  dielectric  give  rise  to  lines  of  magnetic  force  in  a  direction  at 
right  angles  to  the  lines  of  electric  strain  and  the  direction  of 
their  motion. 

(ii.)  Also  that  lines  of  magnetic  flux  moved  laterally  through  a 
dielectric  give  rise  to  lines  of  electric  strain  in  a  direction  at 
right  angles  to  the  lines  of  flux  and  to  the  direction  of  their 
motion. 

The  experimental  proof  of  the  first  statement  has  already  been 
given  by  the  experiments  of  Rowland  and  others  on  the  magnetic 
field  of  moving  electric  charges. 

The  second  statement  when  made  with  regard  to  a  conductor 
is  familiar  to  us  as  Faraday's  Law  of  Induction. 

If  a  bar  of  conducting  material  of  length  L  is  moved  perpen- 
dicularly to  itself  with  a  velocity  F  across  lines  of  magnetic  flux 
of  density  B,  then  we  know  from  Faraday's  law  that  an  electro- 
motive force  (E.M.F.)  is  set  up  in  the  bar  such  that 

E.M.F.=BLV 

reckoned  in  absolute  electromagnetic  units  or  BLV/1Q8  reckoned 
in  volts. 

Now  the  electric  force  E  is  the  electromotive  force  per  centi- 
metre of  length.  Hence  E  =  E.M.F. /L.  Therefore  the  electric 
force  E  set  up  in  the  conductor  is  equal  to  [j.  HV  where  H  is  the 
magnetic  force. 

The  same  will  happen  if  the  conductor  stands  still  and  the 
lines  of  electric  strain  sweep  or  cut  across  it  with  a  velocity  V. 
If  the  bar  is  an  insulator  of  dielectric  constant  K,  then  it  has 


56          PKOPAGATION   OF   ELECTEIC   CURRENTS 

been  shown  theoretically  by  Sir  J.  Larmor  and  experimentally 
by  Professor  H.  A.  Wilson  that  there  is  an  electric  force  set  up  in 
the  bar  when  lines  of  magnetic  force  cut  across  it  with  a  velocity 
V  which  is  expressed  by  the  equation 

HV      .  .     (6) 

This  formula  was  tested  and  verified  by  H.  A.  Wilson  by 
revolving  a  cylinder  of  ebonite  at  a  high  speed  in  a  magnetic 
field  the  lines  of  which  were  parallel  to  the  axis  of  the  cylinder 
around  which  it  revolved.  The  difference  of  potential  between 
the  axis  and  perimeter  was  measured  and  the  mean  electric 
force  equal  to  the  above  difference  of  potential  divided  by  the 
radius  of  the  cylinder  was  calculated  and  found  to  agree  with 
the  above  formula.  For  details  the  reader  is  referred  to  the 
original  paper  (see  Philosophical  Transactions  of  the  Eoyal 
Society  of  London,  Vol.  204A,  p.  121,  1905 ;  also  Proc.  Roy.  Soc., 
Vol.  73,  p.  490,  1904). 

As  regards  the  magnetic  force  produced  by  the  lateral  move- 
ment of  a  line  of  electric  strain,  it  can  be  shown  that  if  E  is 
the  electric  force  in  the  direction  of  the  lines  of  electric  strain 
and  if  K  is  the  dielectric  constant  of  the  medium,  and  if  V  is  the 
velocity  of  the  lines  parallel  to  themselves,  then  the  magnetic 
force  H  produced  by  the  motion  is  given  by  the  formula 

H=KEV (7) 

Otherwise,  if  D  is  the  displacement  or  number  of  lines  of 
electric  strain  passing  through  unit  area,  and  if  they  move  with 
a  velocity  V  in  a  direction  inclined  at  an  angle  0  to  the  direction 
of  the  lines  of  strain,  then  the  magnetic  force  II  due  to  their 
motion  is  given  by 

11=4:  TrDV  Sin  e         ....     (8) 

A  statement  of  the  connection  between  the  electric  force  E 
and  the  magnetic  force  PI  can  be  arrived  at  in  another  way. 
Suppose  we  describe  any  small  area  in  an  electric  field,  say  a 
rectangle  of  which  the  sides  are  dx  and  dy,  and  let  the  electric 
force  E  at  the  centre  of  that  area  have  rectangular  components 
Ex  and  Ey  parallel  to  dx  and  dy  respectively.  Imagine  that  we 
travel  round  the  area  in  a  counter-clockwise  direction,  multi- 
plying the  length  of  each  side  by  the  component  of  the  electric 


ELECTROMAGNETIC   WAVES   ALONG   WIRES        57 

force  in  its  direction  and  reckoning  the  product  as  positive  when 
the  force  is  in  the  direction  of  motion  and  negative  when  it  is 
against  it,  and  finally  add  up  algebraically  all  these  products,  we 
obtain  what  is  called  the  line  integral  of  the  force  round  the  area. 
Thus  for  the  case  in  question  we  have  for  the  line  integral  the 
sum 


*    -      -      -      •    09 

The  above  line  integral  is  the  electromotive  force  acting  round 
the  area,  and  the  quantity  in  the  brackets,  viz.,  —^~    —  ^  is 

called  the  curl  of  the  electric  force  at  that  point  and  written 
Curl  E.  If  there  is  a  magnetic  force  //  in  a  direction  z  at 
right  angles  to  the  plane  of  xy,  then  the  total  magnetic  flux 
through  the  area  fix  by  or  the  number  of  lines  of  electric  force 
passing  through  the  area  is  pH  Sx  by  where  /x  is  the  permeability. 
If  then  the  electromotive  force  is  due  to  the  variation  of  this 
field  we  have  by  Faraday's  law 


or  ur=-f^H        ....     (11) 

where  H  stands  for  —  rr   or  the  time  variation  of  H.     For  the 
at 

above  formula  merely  expresses  the  fact  that  the  electromotive 
force  is  due  to  the  time  rate  of  change  of  the  magnetic  flux 
through  the  area. 

Again,  if  H  is  the  magnetic   force   in   any   field   and  if  its 

rectangular  components  are  Hx  and  Hy  the  quantity  -^--    -~^ 

formed  in  the  same  manner  as  in  the  case  of  the  electric  force 
is  called  the  Curl  of  the  magnetic  force.  If  then  D  is  the 
electric  displacement  normally  through  the  area  Sx  By  drawn  in 
the  magnetic  field,  the  time  rate  of  change  of  this  displacement 

denoted  by  --     or  D  is  called  the  dielectric  current  and  is  the 

C  I  L 


58          PROPAGATION   OF   ELECTRIC   CURRENTS 

rate  at  which  electricity  is  moved  through  the  area.  According 
to  Maxwell's  theory  this  dielectric  current  produces  magnetic 
force  according  to  the  same  laws  as  a  current  of  conduction. 
Hence  4-n-  times  the  total  current  through  the  area  is  equal  to 
the  line  integral  of  magnetic  force  round  the  area.  Applying 
this  to  the  above  case  of  the  dielectric  current  through  the  area 
fix  by  we  have 

(  ~^—y  —  — =-?  ]  &x  8y  =  4-7T — ~bx  by 
\  dx         dy )  dt 

or  CurlH=4irZ> 

or  CurlH=KE     .....     (12) 

The  expressions  therefore  for  the  Curl  of  the  magnetic  force 
and  for  the  Curl  of  the  electric  force  are  quite  similar  and 
involve  the  two  constants  of  the  dielectric,  viz.,  the  magnetic 
permeability  //,  and  the  dielectric  constant  K . 

It  can  be  shown  that  the  velocity  of  propagation  of  any 
electromagnetic  disturbance  or  state  through  a  dielectric  is 
equal  to  1/Vx/L 

For  if  we  consider  that  E  and  H  are  both  at  right  angles  to 
a  common  direction  taken  as  the  #-axis  and  vary  in  that  direction 
alone,  that  is  are  propagated  in  that  direction,  we  have  for  the 
Curl  equations 

^=-Kd^         •         .         •         -     (13) 

^.=  -A^  (14) 

dx         ^  dt 

Hence  differentiating  with  regard  to  x  and  t  we  can  easily  find 
that 


.     (16) 

Now  these  equations  are  precisely  similar  in  form  to  those  we 
deduced  for  the  velocity  of  sound  (see  Equation  (1)  ),  and  they 
show  that  the  velocity  of  an  electromagnetic  disturbance  spreads 

through  the  dielectric  with  a  velocity  «  such  that  u  = 


Thus   if  we   suppose   a  current  in  a  conductor  buried  in  a 
dielectric  to  be  suddenly  reversed  in  direction,  the  magnetic  field 


ELECTROMAGNETIC  WAVES  ALONG  WIRES       59 

due  to  it  is  not  reversed  in  direction  everywhere  at  once,  but  the 
reversal  begins  at  the  surface  of  .the  conductor  and  travels 
outwards  with  a  velocity  1/VKp  where  K  and  ^  are  the  electric 
and  magnetic  constants  of  the  dielectric.  As  regards  numerical 
values  we  do  not  know  the  separate  absolute  values  of  K  and  ^ 
for  air  or  empty  space,  that  is  for  the  tether,  but  we  do  know 
that  the  value  of  the  velocity  u  is  very  nearly  3  X  1010  cms.  per 
second  or  about  1,000  million  feet  a  second — that  is  the  velocity 
of  light. 

Accordingly,  if  lines  of  electric  strain  are  created  at  one  point 
in  a  dielectric  they  diffuse  or  travel  through  it  with  a  velocity  u 
called  the  electromagnetic  velocity,  and  as  they  move  they  give 
rise  to  lines  of  magnetic  flux  at  right  angles  to  themselves  and 
to  their  direction  of  motion.  If  E  is  the  electric  force  and  K  the 
dielectric  constant,  then  the  magnetic  force  H  resulting  from  the 
sidewise  motion  of  the  lines  of  electric  strain  is  given  by 

H=KEu (17) 

Also  if  lines  of  magnetic  flux  move  in  a  similar  manner  the 
electric  force  E  created  is  given  by 

E=fjiHu (18) 

4.  Electromagnetic    Waves    along    Wines. — We 

are  now  in  a  position  to  explain  more  in  detail  the  nature  of  an 
electromagnetic  wave. 

As  we  are  not  concerned  here  with  electric  waves  in  space  or 
so-called  free  or  Hertzian  waves,  but  only  with  waves  guided  along 
wires,  we  shall  take  a  concrete  case,  viz.,  a  pair  of  long  parallel 
wires  of  very  good  conducting  material,  and  examine  the  effects 
taking  place  when  an  electromotive  force  of  particular  type  is 
applied  between  them. 

Let  us  suppose  an  alternator  to  be  applied  at  one  end  giving 
an  electromotive  force  which  rises  suddenly  to  a  certain  value, 
maintains  it  constant  for  a  while,  then  vanishes  and  is  shortly 
afterwards  replaced  by  a  reversed  electromotive  force  going 
through  the  same  cycle  of  values.  The  curve  of  electromotive 
force  or  the  variations  of  E.M.F.  with  time  would  then  be  repre- 
sented by  a  square-shouldered  curve  as  in  Fig.  3. 

If  then  the  E.M.F.  rises  suddenly  at  one  end  of  the  pair  of 


60 


PROPAGATION   OF   ELECTRIC   CURRENTS 


wires  it  implies  that  there  is  an  electric  force  and  therefore  an 
electric  strain  in  the  space  between.  Looking  at  the  wires  end 
on,  the  strain  would  he  distributed  in  curved  lines  as  in  Fig.  4, 


FIG.  3. 

where  the  small  circles  marked  with  a  dot  and  a  cross  represent 
the  section  of  the  wires.  When  looked  at  from  the  side  the  lines  of 
electric  strain  would  project  into  straight  lines  as  in  Fig.  5,  in 
which  the  arrow  heads  represent  the  direction  of  the  electric 

strain.  Now  this  strain 
does  not  make  its  appear- 
ance at  all  distances  at 
once,  hut  is  propagated 
outwards  in  the  space 
between  and  around  the 
wires  at  a  certain  speed, 
and  when  the  electro- 
motive force  at  the  send- 
ing end  dies  down  suddenly 
it  does  not  cease  at  all 
points  at  once.  The  effect 
is  equivalent  to  a  gradual 
movement  of  lines  of  strain 
along  the  space  between 
the  wires.  This  movement 
implies  movement  of  elec- 
tric charges  along  the  wires.  The  ends  of  the  lines  of 
electric  strain,  so  to  speak,  slip  along  the  wires,  and  we  may 
regard  their  ends  as  terminating  on  electric  charges.  But 
this  lateral  movement  of  lines  of  electric  strain  and  of  longi- 
tudinal movement  of  electric  charges  implies  the  flow  of 


FIG.  4. — End-on  view  of  Lines  of  Electric 
and  Magnetic  Forco  of  parallel  wires. 
Firm  lines  are  Magnetic  lines,  Dotted 
lines  are  Electric  lines. 


ELECTEOMAGNETIC   WAVES   ALONG  WIRES        61 

an  electric  current  along  the  wires  and  the  creation  of  lines 
of  magnetic  flux  in  the  interspace,  which  lines  of  flux  are 
everywhere  perpendicular  to  the  lines  of  electric  strain  and  the 
direction  of  the  motion  of  the  latter.  The  lines  of  flux  are  therefore 
closed  loops  embracing  the  wires  as  shown  by  the  dotted  lines  in 
Fig.  4,  and  their  section  is  represented  by  the  dots  in  Fig.  5. 
The  two  distributions  of  lines  of  strain  and  flux  travel  together, 
and  they  both  represent  energy  in  different  forms.  If  the  electric 
strain  density  or  number  of  lines  of  electric  strain  per  square 
centimetre  is  represented  by  D  and  the  number  of  lines  of  mag- 
netic flux  per  square  centimetre  is  represented  by  B,  and  if  the 
dielectric  constant  is  K  and  the  magnetic  permeability  is  //,  then 


FIG.  5.  —  Sidewise  view  of  Lines  of  Electric  and  Magnetic  Force  of 
parallel  wires.  The  arrows  are  electric  lines  and  dots  the 
magnetic  lines. 

the  energy  of  electric  strain  per  cubic  centimetre  is  represented 

1  ID2 

by  0  DE  =  %  j(>  an(i  fcne  energy  of   magnetic  flux  per  cubic 

1  1  .B2 

centimetre  by  ^  HB  ==—,  provided  that  the  flux  and  strain 

lines  are  respectively  practically  parallel  through  the  cubic  centi- 
metre, and  when  B  =  /u  H  and  D  =  KE.  If,  however,  the  values 
of  the  electric  and  magnetic  forces  created  by  the  motion  are 
controlled  by  the  relations  //  =  KEu  and  E  =  ^  Hu,  then  it 
follows  that 


In   other   words,   the   total   energy  is  equally  divided  between 
electric  and  magnetic  forms. 

Hence  as  soon  as  the  lines  of  electric  strain  begin  to  move 


62 


PKOPAGATION   OF   ELECTKIC   CUEBENTS 


freely  they  have  to  part  with  some  energy  or  some  have  to  go  out 
of  existence  to  create  the  lines  of  magnetic  strain,  and  the  total 
energy  is  equally  divided  between  the  two  sets  of  lines. 

If  the  lines  stop  moving,  then  the  magnetic  flux  lines  vanish, 
but  their  energy  cannot  simply  disappear,  but  it  is  conserved  and 
reappears  as  the  energy  of  additional  lines  of  electric  strain 
created.  Conversely  if  the  lines  of  electric  strain  disappear,  then 
their  energy  is  transmitted  into  additional  lines  of  magnetic 
flux. 

Such  a  block  or  group  of  lines  of  electric  strain  travelling 
through  a  dielectric  with  associated  lines  of  magnetic  flux  at  right 
angles  to  the  lines  of  strain,  the  two  groups  being  of  equal 


FIG.  6. 

energy  and  mutually  sustained  by  their  motion,  is  called  an 
electromagnetic  wave. 

Generally  speaking,  in  an  electromagnetic  wave  the  electric 
lines  or  force  do  not  begin  and  end  sharply,  but  fade  away  fore 
and  aft  in  accordance  with  a  sine  law  of  variation,  so  that  it  may 
be  diagrammatically  represented  as  in  Fig.  6,  where  the  close- 
ness of  the  lines  is  supposed  to  denote  the  electric  force  and  of 
the  dots  the  magnetic  force.  We  may  then  mathematically 
express  the  electric  strain  and  magnetic  flux  symbolically  as 

follows  :: — 

D  =  DQSm(x-Vt)     ....     (19) 

B  =  B08m(x-Vt)      ....     (20) 

where  D0  and  -Bo  represent  the  maximum  values  of  the  electric 
strain  and  magnetic  flux  and  D  and  B  their  values  at  any 
distance  x  from  an  origin  and  any  time  t,  and  V  is  the  velocity 
of  propagation.  For  these  expressions  are  periodic  both  in 
space  and  time  and  remain  the  same  if  for  x  we  put  x  +  ^  and 


ELECTROMAGNETIC   WAVES   ALONG   WIRES        63 

for  t,  t  +  T,  provided  \/T  =  F.  The  length  A  is  called  the  wave 
length  and  the  time  T  is  called  the  periodic  time.  The  former 
is  the  length  in  which  the  whole  cycle  or  series  of  electric  or 
magnetic  lines  is  contained  at  any  instant,  and  the  time  T  is 
the  time  in  which  the  whole  cycle  of  variations  completes  itself 
at  any  one  place.  If  then  we  have  an  ordinary  alternator  attached 
to  the  end  of  the  line,  producing  a  simple  harmonic  electromotive 
force,  we  have  sinoidal  electromagnetic  waves  travelling  up  the 
space  between  the  wires  with  a  velocity  V  presently  to  be  deter- 
mined and  constituting  a  train  of  electromagnetic  waves.  In  a 
pure  electromagnetic  wave  the  energy  is  half  electric  and  half 
magnetic,  and  the  two  constituents,  the  magnetic  component  and 
the  electric  component,  travel  together  with  the  same  speed  and 
are  in  step  as  regards  phase.  As  regards  the  relative  direction 
of  the  electric  force  or  strain,  magnetic  force  or  flux,  and  motion, 
their  direction  can  be  remembered  by  holding  the  thumb,  middle 
finger,  and  fore  finger  of  the  right  hand  in  directions  as  nearly  as 
possible  at  right  angles.  Let  the  direction  in  which  the  thumb 
points  indicate  the  direction  of  the  wave  motion  or  velocity,  the 
direction  in  which  the  middle  finger  points  the  direction  of  the 
magnetic  force  or  flux,  and  the  direction  in  which  the  fore  finger 
points  the  direction  of  the  electric  force  or  strain.  Then  by 
twisting  round  the  hand  into  various  directions  with  the  thumb, 
and  two  fingers  held  stiffly  at  right  angles,  we  can  always  deter- 
mine the  directions  of  the  magnetic  and  electric  vectors,  as  they 
are  termed,  with  regard  to  the  direction  of  wave  propagation. 

5.   Reflection    of   Electromagnetic    Waves    at 

the  End  of  a  Line.— Before  proceeding  to  discuss  analy- 
tically the  propagation  of  waves  along  wires  it  will  be  found 
profitable  to  consider  the  phenomena  which  occur  when  an 
electromagnetic  wave  reaches  the  end  of  a  line  whether  open  or 
closed. 

First  consider  an  open  or  insulated  end.  When  the  lines 
of  strain  arrive  at  the  end  of  the  line  they  cannot  proceed 
farther  because  their  ends  cannot  be  detached  from  the  metal 
wires,  but  their  inertia  causes  them  to  travel  as  far  as  they  can 
by  stretching  themselves ;  hence  as  they  reach  the  end  of  the 


64          PEOPAGATION   OF   ELECTRIC   CURRENTS 

line  they  extend  themselves  in  curved  lines  as  shown  in  Fig.  7. 
As  soon,  however,  as  they  come  to  rest,  the  accompanying 
magnetic  flux,  which  is  produced  only  by  the  sidewise  motion  of 
the  strain  lines,  vanishes,  but  its  disappearance  results  in  the 
creation  of  additional  lines  of  electric  strain  to  conserve  the 


FIG.  7. 

energy.  Some  of  the  electric  strain  lines  are  then  in  a  state  of 
stretch,  but  owing  to  their  longitudinal  tension  they  tend  to  con- 
tract and  to  start  the  whole  mass  of  strain  lines  back  again  on 
the  return  journey.  As  soon,  however,  as  the  lines  begin  to 
travel  their  motion  recreates  the  magnetic  flux  lines,  and  part  of 
the  electric  strain  lines  vanish  to  supply  the  magnetic  energy. 
Then  the  wave  is  re-established  and  runs  back  again  to  the 
origin.  Here  it  may  be  reflected  again  and  so  travel  backwards 
and  forwards  until  its  energy  is  dissipated.  If  the  receiving  end 


FIG.  8. 

of  the  cable  is  short  circuited  by  a  good  conductor,  then  the 
process  of  reflection  is  somewhat  different.  When  the  strain 
lines  arrive  at  the  end  their  ends  follow  on  round  the  short 
circuit,  and  the  strain  lines  therefore  tend  to  shrivel  up  to 
nothing,  as  shown  in  Fig.  8.  But  this  process  implies  a  move- 
ment of  each  part  of  the  strain  line  at  right  angles  to  itself  and 


ELECTROMAGNETIC   WAVES  ALONG   WIRES       65 

so  gives  rise  to  a  line  of  magnetic  flux  embracing  the  end  con- 
ductor, which  is  left  behind  as  the  equivalent  form  of  energy 
when  the  electric  strain  lines  disappear.  Hence  when  the  wave 
reaches  the  closed  circuit  end  all  the  lines  of  electric  strain 
vanish  for  the  moment  and  are  replaced  by  lines  of  magnetic 
flux.  But  this  state  is  not  stable.  The  closed  lines  of  magnetic 
flux  closely  embracing  the  short  circuit  end  begin  to  expand 
outwards  again  like  ripples  on  a  pond,  and  the  moment  they 
move  their  motion  recreates  electric  strain  lines,  and  presently 
the  energy  is  again  divided  equally  between  electric  strain  and 
magnetic  flux  lines  in  lateral  motion. 

Accordingly  we  see  that  there  are  two  general  laws  as  follows. 

1.  When  an  electromagnetic  wave  is  reflected  at  the  open  end 
of  a  cable  the  magnetic  component  is  reversed  in  direction,  and 
at  the  moment  of  reflection  the  magnetic  component  is  suppressed 
and  the  electric  component  doubled  in  intensity. 

2.  WThen  an  electromagnetic  wave   is   reflected  at  the   short 
circuited  end  of  a  cable  the  electric  component  is  reversed  on 
reflection,  and  at  the  moment  of  reflection  the  electric  component 
is  suppressed  and  the  magnetic  component  doubled  in  intensity. 
If   the   electromagnetic   waves   continue   to   arrive   and    to    be 
reflected  at  the  open  or  closed  end,  then  the  two  trains  of  waves, 
direct  and  reflected,  pass  through  each  other,  and  the  resultant 
state  of  affairs  is  said  to  be  due  to  the  interference  of  the  direct 
and  reflected  wave  trains. 

If  the  receiving  end  is  not  perfectly  insulated  or  perfectly 
short  circuited  the  energy  of  the  wave  is  partly  reflected  and 
partly  transmitted  and  the  resulting  condition  becomes  still  more 
complicated. 

We  may  make  an  additional  inference.  If  there  be  in  any 
cable  a  part  in  which  there  is  greater  inductance  or  capacity  per 
unit  of  length  than  at  other  parts,  these  lumps  of  capacity  or 
inductance  will  cause  partial  reflection  of  the  wave. 

The  whole  process  of  transmission  and  reflection  of  electro- 
magnetic waves  up  the  space  between  two  conducting  wires  is 
exactly  analogous  to  the  phenomena  occurring  when  air  waves 
are  travelling  up  a  pipe  such  as  an  organ  pipe. 

In  place  of  magnetic  flux  we  have  to  consider  the  velocity  of 

B.C.  P 


66    PROPAGATION  OF  ELECTRIC  CURRENTS 

the  air  particles,  and  in  place  of  electric  strain  we  have  air  con- 
densation or  rarefaction.  If  the  pipe  is  closed  at  the  end  then 
when  the  air  wave  reaches  it,  it  is  reflected  with  change  of  the 
direction  of  the  velocity  component.  If  the  pipe  is  open  at  the 
end  the  wave  is  also  reflected,  but  with  change  of  phase  of  the 
density  component,  that  is  a  condensation  is  reflected  as  a 
rarefaction  and  vice  versa. 

In  the  air  wave  at  any  one  point  changes  of  density  succeed 
each  other  periodically,  and  also  changes  in  the  velocity  of  the 
air  particles.  In  the  electromagnetic  wave  changes  of  electric 
strain  and  electric  force  in  amount  and  direction  succeed  each 
other  at  any  one  point,  and  also  similar  changes  in  magnetic  flux 
or  force. 

If  the  wave  could  be  arrested  and  fixed  in  its  state  at  any  one 
moment  we  should  find  a  periodic  distribution  of  electric  and 
magnetic  force  in  space,  the  two  being  in  mutually  perpendicular 
directions  and  also  at  right  angles  to  the  direction  of 
propagation. 

6.  Differential  Equations  expressing  the  Pro- 
pagation of  an  Electromagnetic  Disturbance 
along  a  pair  of  Wires. — Having  obtained  a  general  con- 
ception of  the  nature  of  the  physical  processes  taking  place  when 
a  simple  periodic  electromotive  force  is  applied  to  a  pair  of 
parallel  wires,  we  shall  next  proceed  to  translate  these  ideas  into 
mathematical  language  in  order  to  give  greater  precision  to 
them. 

Let  us  consider  a  transmission  line  consisting  of  two  parallel 
infinitely  long  wires  having  a  resistance  R  ohms  per  mile  of 
line,  that  is  per  mile  of  lead  and  return,  and  a  capacity  C  farads 
per  mile,  an  inductance  L  henrys  per  mile,  and  a  dielectric 
.conductance  of  S  mhos  per  mile,  the  mho  being  the  reciprocal  of 
the  ohm. 

Let  v  be  'the  potential  difference  between  the  wires  at  any 
distance  x  from  the  sending  end  and  let  i  be  the  current  at  that 
point.  Then  the  potential  'difference  and  current  at  a  distance 

is  v + &r and?  +,&r- 


ELECTROMAGNETIC   WAVES   ALONG   WIRES       67 

The  potential  difference  (P.  /).)  is  partly  expended  in  driving 
the  current  against  the  ohrnic  resistance  and  partly  in  over- 
coming the  back  electromotive  force  due  to  inductance.  Hence 
for  a  length  bjc  we  have  the  following  equations. 

^8x  =  R8xi+Ux^    .  (21) 

dx  dt 


-  .  (22) 

dx  dt 

The  first  equation  expresses  the  manner  in  which  the  fall  in 
voltage  down  the  length  bx  is  accounted  for,  and  the  second  the 
manner  in  which  the  current  in  the  same  length  is  expended 
partly  in  charging  the  wire  and  partly  in  conduction  across  the 
dielectric.  From  these  equations  we  at  once  derive  the 
following  :— 

l=B*+i|         -...     (23) 


These  are  the  differential  equations  for  the  propagation  of  a 
current  in  a  line  having  resistance,  inductance,  capacity,  and 
insulation  conductance.  W7e  need  not  consider  at  the  present 
moment  the  general  solution  of  these  equations,  but  for  the 
immediate  purposes  we  shall  limit  our  consideration  to  the  case 
in  which  both  v  and  i  are  simple  sine  functions  of  the  time. 
Then  if  i  —  I  Sin  pt  and  v  =  V  Sin  (pt  +  0),  these  functions 
indicate  a  simple  sine  variation  of  i  and  v  with  a  difference  of 
phase  0  but  equal  frequency  n  such  that  %im=p.  Thus  we  have 

Hence    if    we    denote    the 


periodic  current  by  a  simple  vector  representing  its  maximum 
value,  then  a  vector  p  times  as  long  at  right  angles  to  the  vector 
denoting  the  current  will  represent  the  maximum  value  of 
the  time  rate  of  change  of  the  current  or  the  maximum  value 
tdi 
oidt 

If  therefore  any  line  is  taken  to  represent  RI,  or  the  maximum 

value  of  Rif  then  for  the  maximum  value  of  L  -=-  we  must  draw 

dt 

a  line  to  the  same  scale  representing  Lpl  at  right  angles  to  the 

F  2 


68         PBOPAGATION   OF   ELECTEIC   CUERENTS 

vector  RL     The  vector  sum  of  these  lines  or  RI+jpLI  will  be 

dV 
a  line  representing  the  maximum  value  of  ^—  .     Hence    when 

the  time  variation  of  i  and  v  are  simple  harmonic,  we  can,  in  place 
of  the  scalar  equation 

dv  ^  di 

dx 
write  the  vector  equation 

dV 


where  V  and  /  are  the  maximum  values  of  v  and  i  during  the 
period.  We  thus  eliminate  the  time  variable  and  deal  only  with 
the  maximum  values  of  the  quantities  during  the  period. 

Hence  we  can  write  our  two  fundamental  equations  in  the 
form, 

^=(R+jpL)I      .  .    (25) 

=(S+jpC)V       .  .        .    (26) 


The  quantity,/?  +  jpL  is  called  the  rector  impedance,  and  the 
quantity  S  +  jpC  is  called  the  rector  admittance. 

By  differentiating  each  of  the  equations  above  with  regard  to  x 
we  can  separate  the  variables  and  arrive  at  the  two  equations, 

.    (27) 

.     (28) 

where  P=  jR+jpL      s+j^C=a+jp  .        .        .     (29) 

P  is  a  complex  quantity  and  therefore  may  be  written  in  the 
form  a  +  jfi. 

It  is  called  the  Propagation  constant  of  the  line. 
Squaring  the  two  sides  of  the  expression  (29)  we  have 

a2-/32+y  2  ap  =  (RS-p*LC)+j  (pLS+p  CR\ 
and  equating  horizontal  and  vertical  steps  we  have, 

a*-(32=RS-p*LC    ....     (30) 
and  Za$=p(LS  +  CR)     ....    (31) 


ELECTROMAGNETIC   WAVES   ALONG   WIRES        69 
and  equating  the  sizes  of  the  vectors  we  have 


or,  a*+p2=^+P'2L*(SP+p**)        •        -     (32) 

whence  we  find  that 


=  \J 


C*)  +  (SR-p*LC)       •     (33) 


v  -        -  (34) 

These  quantities  a  and  /3  are  very  important,  a  is  called  the 
attenuation  constant,  and  /3  is  called  the  wave  length  constant,  and 
P  =  a  -\-jj3  is  the  Propagation  constant  of  the  line. 

The  expressions  for  aand'/3  may  be  modified  by  relative  values 
of  7i,  L,  $  and  C,  which  last  are  called  the  primary  constants  of 
the  line,  a,  8,  and  P  being  the  secondary  constants. 

Thus  if  £  =  0  or  the  line  has  no  leakance,  then 


.    (35) 


.        .        .     (36) 
If  S  =  0  and  L  =  0  or  the  line  has  no  inductance  as  well,  then 


In  all  these  cases  a  and  /3  are  functions  of  p  and  therefore  of 
the  frequency  n,  since  p  =  27m. 

In  the  general  expressions  for  a  and  /3,  viz.,  in  equations 
(33)  and  (34),  if  we  add  and  subtract  the  quantity  ZjPCLSR 
to  and  from  the  product  (fl2+^2L2)  (S2+p2  C2)  we  can  throw 
the  expressions  for  a  and  /3  in  the  form 


-^  LC)       -     (38) 
•     (39) 


If  tben  the  primary  constants  have  such  values  that 

LS-CB  =  QorLS  =  CB, 
then  we  have 

a=  JSR       .  -     (40) 

.....     (41) 


70          PROPAGATION   OF   ELECTRIC   CURRENTS 

For  a  reason  to  be  explained  later  on,  such  a  cable  is  called  a 
distorsionless  cable,  and  in  that  case  the  value  of  a  is  the  same  for 
all  frequencies. 

We  have  therefore  obtained  differential  equations  expressing 
the  relations  of  the  potential  and  current  at  any  point  in  a  line 
under  the  assumption  that  the  current  and  potential  are  quantities 
which  vary  in  accordance  with  a  simple  sine  law. 

In  the  next  chapter  we  shall  discuss  the  solution  of  these  simple 
periodic  equations  in  various  cases,  and  deal  with  the  general 
solution  of  the  differential  equations  (23)  and  (24)  at  a  later 
stage. 


CHAPTEE  III 

THE    PROPAGATION    OF    SIMPLE    PERIODIC    ELECTRIC    CURRENTS    IN 
TELEPHONE    CABLES 

1.  The  Case  of  an  Infinitely  Long  Cable  with 
Simple  Periodic  EEect  no  motive  Force  at  the 
Sending  End.—  Eeturning  to  the  fundamental  differential 
equations  we  have  now  to  find  solutions  for  particular  cases. 
These  equations  are, 

•  •  •  .  •  .  « 


where  V  and  /  are  the  maximum  values  during  the  period  of  the 
potential  and  current  at  any  point  in  the  line.  A  differential 
equation  of  this  type  can  be  satisfied  by  simple  exponential 
solutions  of  the  form  V  =  Ac+Px  and  V  =  B*~PX  where  .4  and  B 
are  constants,  as  can  be  seen  at  once  by  double  differentiation 
of  these  last  expressions.  Hence  a  solution  of  these  equations 
is  found  by  taking  the  sum  of  the  above  particular  solutions, 
viz., 

''x        .         .  .     (3) 

1'*  .  .      (4) 

where  A,  B,  C,  and  D  are  constants  to  be  determined. 

In  obtaining  the  original  differential  equations  (23)  and  (24) 
(see  Chapter  II.)  it  is  to  be  noticed  that  we  assumed  the  current 
and  potential  to  increase  with  x.  It  is  most  convenient  to  reckon 
the  distance  x  from  the  sending  end,  and  then  V  and  /  both 
diminish  with  x.  We  can,  however,  make  the  necessary  change 
in  our  solutions  by  writing  —  x  for  x.  Making  the  change,  we 
have  for  the  solutions  of  (1)  and  (2). 

*         ....     (5) 
v         ....     (6) 


72          PEOPAGATION   OF   ELECTRIC   CURRENTS 

Suppose  that  the  cable  is  infinitely  long,  then  when  x  =  oo  we 
must  have  V  —  0  and  I—  0,  and  therefore  the  constants  B  and  D 
must  both  be  zero.  Again  when  x  =  0  we  must  have  V  =  E0 
where  EQ  is  the  potential  difference  of  the  two  members  of  the 
cable  at  the  origin,  or,  in  other  words,  the  electromotive  force 
applied  at  the  sending  end.  Hence  A  =  EQ.  Moreover,  since 

)  I,  it  follows  that 


dx 

K  or  C= 


Hence  in  the  case  of  the  infinitely  long  cable  with  simple 
periodic  E.M.F.  E0  applied  at  the  sending  end,  the  potential  and 
current  at  any  distance  x  are  given  by  the  equations, 

V=E&->'*      •  •  •  •      (7) 


(r- 
JR+jpL 


where  P=  jR~+jpL  jS+jpU~=0.+jp. 

The  quantity  P  is  called  the  propagation  constant  and  Px  is  called 

the  propagation  length  or  distance. 

The  quantity  ~-^    .  ^    is  of  great  importance  and  is  called 

the  Initial  sending  end  Impedance  and  denoted  by  ZQ. 

Bearing  in  mind  that  c~^  =  Cos  fix  —  j  Sin  fix  and  that 
€~PX  =  c-M-Jfta  it  is  seen  that  the  solutions  of  the  differential 
equations  for  the  case  of  the  infinitely  long  cable  can  be  put  in 
the  form 

V=  E0  e-«*  (Cos  PX  -j  Sin  /Ete)      .  '     '  .         .     (9) 


jo  €-ax  (Cos  PX-J  Sin  PX)       . 

^0 

Each  of  these  expressions  is  a  complex  quantity  and  therefore 
represents  a  vector.  The  value  of  V  is  obtained  by  operating  on 
EQ  with  two  factors ;  one  viz.,  €~ax,  called  the  attenuation  factor, 
continually  decreases  in  a  geometric  progression  as  x  increases  in 
an  arithmetic  progression.  The  other  factor  (Cos  fix  —j  Sin  fix), 
called  the  phase  factor,  repeats  itself  over  and  over  again  in  value 

at  intervals  of  distance  equal  to  -Q    as  x  continually  increases, 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  73 

since  Cos  fix  =  Cos  /3  (x^~~i~   and  the  same  for  the  sine  term. 


It  is  clear,  therefore,  that  as  we  move  along  the  line  the 
potential  and  current  rise  and  fall  periodically,  but  so  that  the 
maximum  value  in  each  space  period  dies  gradually  away. 
Moreover  at  each  point  the  potential  and  current  are  periodic 
with  time  ;  that  is,  run  through  a  cycle  of  values. 

This  shows  that  as  we  proceed  along  the  cable,  taking  the 
potential  and  current  at  each  point  to  be  the  maximum  values 
they  have  during  the  period,  we  find  that  these  maximum  values 
attenuate  in  a  certain  ratio  and  are  shifted  backwards  in  phase 
relatively  to  each  other.  At  equal  space  intervals  along  the  line 


FIG.  1. 

these  maximum  values  form  a  geometric  series  as  regards  their 
size  and  their  phases  differ  by  equal  angles.     The  distance  ~ 

is  called  the  wave  length  and  denoted  by  A. 

We  can  represent  the  state  of  affairs  in  the  cable  by  a  model 
made  in  the  following  manner  : 

Take  a  long  wooden  rod  to  represent  the  cable  and  a  number 
of  wires  the  lengths  of  which  form  a  geometric  series,  that  is  the 
length  of  each  wire  is  the  same  fraction  or  percentage  of  the 
next  longest  one.  Let  holes  be  bored  in  the  wooden  rod  at  equal 
distances  and  in  such  directions  that  these  holes  lie  on  a  spiral 
of  equal  pitch  wound  round  the  rod,  the  holes  being  otherwise 
perpendicular  to  the  axis. 

Then  if  the  wires  are  inserted  into  the  holes  we  shall  have  a 
structure  as  shown  in  Fig.  1. 

Each  wire  will  then  represent  in  magnitude  and  direction  the 
maximum  value  and  phase  of  the  current  or  potential  at  the 


74         PEOPAGATION   OF   ELECTRIC   CUREENTS 

corresponding  point  in  the  cable.  If  the  tips  of  all  these  wires 
are  joined  by  another  wire,  this  last  will  form  a  spiral  round  the 
rod,  but  the  spiral  will  be  like  a  corkscrew,  decreasing  in  diameter 
the  further  we  move  along  the  rod.  If  we  wish  to  represent  the 
changes  which  take  place  from  instant  to  instant  in  the  potential 
or  current  we  must  place  this  rod  in  the  sunshine  and  cast  the 
shadow  of  it  on  a  sheet  of  white  paper  held  perpendicular  to  the 
sun's  rays.  If  then  the  rod  is  rotated  the  shadow  of  each  of  the 
wires  will  increase  and  decrease  and  reverse  direction  at  each 
turn.  The  length  of  the  shadow  at  any  instant  will  denote  the 
actual  current  or  potential  at  that  point  in  the  cable  and  runs 
through  a  cycle  of  values  at  each  revolution  of  the  rod. 


FIG.  2. 

A  line  joining  the  tips  of  all  the  shadows  will  at  any  moment 
be  a  wavy  decrescent  curve  as  in  Fig.  2,  and  as  the  rod  is  rotated 
the  ends  of  these  shadow  lines  will  appear  to  move  forward  with 
a  wavy  motion. 

The  curve  formed  by  joining  the  tips  of  the  shadow  lines  is  a 
curve  like  that  in  Fig.  2  whose  equation  is  of  the  form 

y=A  €~ax  Cos  j8a?  (11) 

Hence  if  we  suppose  ourselves  to  stay  permanently  at  a  point 
in  the  cable  the  distance  of  which  from  the  sending  end  is  x, 
we  should  find  the  potential  and  current  at  that  point  varying 
periodically  with  a  frequency  n  or  having  a  periodic  time  T. 

If  we  could  cause  the  current  and  potential  at  all  points  in 
the  line  to  be  fixed  permanently  in  the  state  in  which  they  are 
at  any  instant  t,  then  we  should  find  a  distribution  along  the 
line  which  is  periodic  with  a  wave  length  2  ir/p,  but  the 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  75 
maximum  values  in  each  half  wave  length  decreasing  in  the 

7T 

ratio  e~a/3. 

A  model  imitating  the  actual  changes  of  potential  from  instant 
to  instant  at  any  point  in  the  cable  can  be  made  in  the  following 
manner. 

On  a  long  axis  are  fixed  a  series  of  grooved  eccentric  wheels 
the  eccentricities  of  which  decrease  in  geometric  progression — 
that  is  the  eccentricity  of  each  one  is  the  same  fraction  of  that 
of  the  preceding  one  all  the  way  along.  Also  the  angle  of  lead 
of  these  wheels  decreases  progressively  by  equal  angular  steps. 
In  each  wheel  is  a  groove  on  which  is  hung  a  long  loop  of 
string  carrying  a  weight  at  the  bottom.  The  loops  are  all  of 
equal  length.  These  weights  therefore  are  arranged  along  a 
wavy  decrescent  curve.  If  the  axis  is  rotated  each  bob  moves  up 
and  down  with  a  nearly  simple  harmonic  motion,  but  the 
amplitudes  of  motion  decay  in  a  geometric  progression  and 
the  phases  lag  in  arithmetric  progression,  and  hence  the  bob 
motion  represents  in  phase  and  amplitude  the  potential  or 
current  at  various  points  along  the  cable. 

A  model  of  this  kind  has  actually  been  constructed  by  the 
author  and  exhibited  in  various  places.1 

If  then  for  any  cable  we  are  given  the  primary  constants 
R,  L,  C,  S,  in  ohms,  henrys,  farads,  and  mhos,  per  mile,  we  can 
calculate  the  values  of  the  attenuation  constant  a  and  the  wave 
length  constant  (B  and  hence  the  attenuation  factor  €~ax  and  the 
phase  factor  Cos  fix—  j  Sin  fix  for  any  distance  x.  The  attenua- 
tion per  mile,  viz.,  e~a,  and  the  wave  length  2-/T//3  are  then  at 
once  found. 

The  value  of  t~ax  can  be  calculated  most  easily  by  means  of 
a  Table  of  Hyperbolic  Sines  and  Cosines. 

For  €""•*=:  Cosh  ax  —  Sinh  ax. 

Hence  V=E()  (Cosh  ax- Sinh  ax) (Cos  fix-j  Sin  fix)        .     (12) 

J=§Q  (Cosh  cur -Sinh  our)(Cos  ftx-j  Sin  fix)        .     (13) 

^o 

'See  "A  Model  illustrating  the  Propagation  of  a  Periodic  Current  in  a  Tele- 
phone (.'able  and  the  Simple  Theory  of  its  operation,"  Phil.  Mr(;/..  August,  1904, 
an.  I  l'rin:  /'////*.  tin:  Loud.,  Vol.  XIX. 


76          PROPAGATION   OF   ELECTRIC   CURRENTS 

If  we  reckon  phase  angles  from  the  direction  of  EQ,  then 
symbolically  we  have 

V=  (E0)  (Cosh  ax  -  Sinh  ax)\fix  .        • .         .     (14) 

J=  (?-()VCosh  ax  ~  Sinh  ax)\Bx.  .     (15) 

\/40/ 

where  the  brackets  round  EQ  and  -^-   denote  the  sizes  of  these 

vectors. 

We  have  therefore  completely  determined  the  potential  and 
current  at  any  point  in  the  infinite  cable. 

Moreover,  given  the  values  of  It,  L,  C,  and  S,  per  mile  or  per 
kilometre,  we  can  calculate  the  value  of  the  attenuation  constant 
a  and  hence  of  e~a,  which  gives  us  the  attenuation  per  mile  or 
ratio  in  which  the  maximum  values  of  the  current  and  potential 
are  weakened  by  going  a  mile  or  kilometre  along  the  cable. 
Also  we  can  calculate  the  value  of  the  wave  length  constant, 
which  the  formula  (34),  Chapter  II.,  gives  in  radians,  the  radian 
being  the  unit  angle  or  angle  whose  arc  is  equal  to  the  radius, 

180 
viz.,    :  -=57°   17'   45"   nearly.     Accordingly  the   angle  of   the 

7T 

vector  denoting  the  current  or  potential  is  shifted  backwards  by 
P  —  degrees  per  mile  or  per  kilometre. 

Hence  after   running  a  distance  —  the   phase   has    shifted 

backwards  360°  and  the  cycle  as  regards  phase  begins  again  to 
be  repeated.     The  length  2ir//3  =  A  is  called  the  wave  length. 

Now   in   all  cases  of  wave  motion   the  wave  velocity   W  is 
connected  with  the  wave  length  A  and  the  frequency  u  in  the 
L^    relation  giv7en  by  the  formula 

W=nK       .        ....     (16) 
But  A  =  27T//3  and  27m  =  p,  and  hence 

W=P  (17) 

A 

s\ 

Accordingly   the    velocity   of   the   wave   is    a   function   of    the 
frequency  n. 

It  is  therefore  seen  that  in  an  ordinary  cable  alternating 
currents  or  potentials  of  different  frequency  decay  at  different 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES   77 

rates  along  the  cable  and  travel  with  different  velocities.  There 
is,  however,  one  important  case  in  which  currents  of  all  fre- 
quencies attenuate  equally  and  travel  at  the  same  speed.  This 
is  when  the  primary  constants  have  such  values  that 


We  have  seen  that  under  these  conditions 

and  = 


Hence  W=      ==.     When  this  is  the   case   both  a  and   W  are 

\/  (JL 

independent  of  the  frequency,  and  currents  and  potentials  of  all 
frequencies  travel  and  attenuate  alike. 

Such  a  cable  has  been  called  by  Mr.  Oliver  Heaviside  a 
distortionless  cable,  for  reasons  to  be  considered  later  on. 

In  the  case  of  all  ordinary  cables  the  values  of  the  constants 
are  such  that  the  product  R  C  is  much  greater  than  the  product 
L  S.  It  is  easily  seen  that  under  these  conditions  the  lower  the 
frequency  the  less  the  attenuation  and  wave  velocity  but  the 
greater  the  wave  length.  Hence  shorter  waves  travel  faster  and 
attenuate  more  rapidly. 

Thus  for  instance  take  the  cable  to  be  the  National  Telephone 
Company's  Standard  Telephone  Cable,  which  has  the  following 
constants  per  loop  mile,  that  is  per  mile  run  of  lead  and  return, 
R  =  88  ohms,  C  =  '05  microfarad,  L  =  '001  henry,  S  =  0. 

Suppose  we  apply  a  simple  periodic  E.M.F.  at  one  end  of 
such  a  cable  infinite  or  very  great  in  length. 

Let  the  frequency  n  be  83,  which  gives  p  =  £&vn  =  500  nearly. 

We  have  then 

i  05 

LP=~,  cp=w, 

25 


Hence 


or  a=  -034,    |8  = 

and  A  =       =  185  miles 


o,2  12-5 

2/3  -o6 


=    =  15,000  miles  per  second. 
p 


78          PROPAGATION   OF   ELECTRIC   CURRENTS 

Next  suppose  n  =  830  or  p  =  5,000. 

Then  Lp  =  5,  Cp  =  ~^,  VB*+p*L*  =  QQ'l 


88         1 
4000"  §00     a  =  '104 


Hence  A  =  62'8  miles,  W  =  50,000  miles  per  second.  Finally, 
if  n  =  8,300,  and  p  =  50,000,  we  find  that  a  =  -253,  /3  =  -435, 
and  \  =  16  miles,  W  =  125,000  miles  per  second. 

This  cable  is  therefore  very  far  from  being  distorsionless. 

As  the  frequency  continually  increases  the  wave  velocity 
approximates  to  the  velocity  of  light,  viz.,  186,000  miles  per 
second,  which,  however,  it  can  never  exceed. 

2.  Propagation  of  Simple  Periodic  Currents 
along  a  Cable  of  Finite  Length.—  We  have  next  to 
consider  the  modifications  produced  in  the  above  formulae  when 
the  cable  is  finite  in  length.  This  is  the  case  which  presents 
itself  in  practice.  We  then  find  that  the  reflection  of  the  current 
or  potential  wave  at  the  receiving  and  sending  ends  of  the  cable 
introduces  considerable  modifications  into  the  above  formulae. 

Returning  to  the  general  expressions  for  the  potential  and 
current  at  all  points  in  an  infinite  cable,  viz., 

K    ....   (19) 


Let  us  write  for  e~^,  Cosh  Px  -  Sinh  Px,  and  for  e+p*, 
Cosh  Px  +  Sinh  Px,  and  rearrange  terms.  We  then  transform 
the  above  equations  into 

7=  (A  +  B)  Cosh  Px  -  (A  -  B)  Sinh  Px      .         .     (21) 

1=  ~  |  (A  -  B)  (Cosh  Px  -  (A  +  B)  Sinh  Px  1       .     (22) 

Now  if  x  =  0,  Sinh  Px  =  0,  and  Cosh  Px  =  1,  and  if  we 
call  FI  and  Ii  the  potential  difference  and  current  at  the  sending 
end,  then  when  x  =  0,  we  find  that 

(A  +  B)=  V,  and  (A-B)  =  ll  ZQ 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  79 

Suppose  that  the  potential  and  current  at  the  receiving  end 
denoted  by  F2  and  72  and  that  the  cable  has  a  length  /.  Then 
at  a  distance  x  from  the  sending  end  and  l  —  x  from  the  receiving 
end,  if  the  potential  and  current  are  V  and  I,  we  can  write  the 
expressions  for  V  and  I  in  two  forms,  viz., 

F=  F!  Cosh  Px-I,  Z()  Sinh  Px  .         .         .         .     (23) 
=  F2  Cosh  P(l  -  x)  +  12  ZQ  Sinh  P(l  -x)     .         .     (24) 

1=1,  Cosh  Pz  -  -5  Sinh  Pa;  .  .     (25) 

^o 

(/-£)+-5  Sinh  P(Z-a)  .     (26) 


The  equations  (23)  and  (25)  are  obtained  from  (21)  and  (22) 
by  substituting  V\  for  A  4-  B  and  /i  Z0  for  ^.  --  B. 

The  equations  (24)  and  (26)  are  obtained  by  reckoning  the 
distance  from  the  receiving  end  and  assuming  the  voltage  and 
current  at  that  end  to  correspond  to  x  =  0.  The  signs  are 
changed  because  in  the  last  case  the  current  and  voltage  increase 
along  the  cable  with  distance  reckoned  from  the  receiving  end. 

The  above  equations  ^23)  and  (25)  give  the  complete  solution 
of  the  problem  for  the  case  of  a  finite  cable,  and  we  have  three 
cases  to  consider,  viz.,  (i.)  when  the  receiving  end  is  free  or 
insulated,  (ii.)  when  the  receiving  end  is  short  circuited,  and  (iii.) 
when  it  is  closed  by  a  receiving  instrument  of  known 
impedance. 

3.  Propagation  of  Currents  along  a  Finite  Cable 
Free  or  Insulated  at  the  Receiving  End.—  In  this 
case  the  current  L2  must  be  zero.  Hence  in  the  general 
equations  corresponding  to  x  =  I  we  must  have  I  —  0,  and 
making  this  substitution  in  equation  (25)  this  gives  us, 

0  =  Il  Cosh  Pl-^  Sinh  PI     .         .        .     (27) 
^o 

or  I1^0=F1TanhPZ      .  .     (28) 

Substituting  this  value  for  Ii  Z0  in  (23)  we  have 

V=  Vl  [Cosh  Px  -  Tanh  PI  Sinh  Px]         .         .     (29) 
This  equation  gives  us  the  potential  difference  F  (maximum 


80         PROPAGATION   OF   ELECTEIC   CURRENTS 

value)    at  any  distance  x  along  a  cable  Laving  a   Propagation 
Constant  P  which  is  open  at  the  far  end. 
Again  from  (28)  we  have 

¥?  =  Zi  =  Z  nCothP/  (30) 

A 

The  ratio  of  the  applied  voltage  to  the  current  at  the  sending 
end  is  called  the  final  sending-cnd  impedance  and  denoted  by  Z\. 
The  reader  should  carefully  distinguish  between  the  final  sending- 
end  impedance  Zi  =  Vi/Ii  and  the  initial  sending -end  impe- 
dance z0  =  VR  +jpL/Vs+jpC. 

If  we  compare  the  above  expressions  for  V  and  Vi/Ii  for  the 
finite  cable  with  the  corresponding  expressions  for  the  infinite 
cable  the  reader  will  at  once  see  how  the  hyperbolic  expressions 
are  modified  when  there  is  reflection  at  the  ends  of  the  cable. 
For  in  the  case  of  the  infinite  cable  we  have  seen  that 

V=  V,  f-r*  =  Vl  [Cosh  Px  -  Sinh  Px]        .         .     (31) 

and  TI  =  ^O 

J-i 

whilst  for  the  finite  cable  of  length  I  we  have 

V=  V,  [Cosh  Pz-Tauh  PI  Sinh  Px]        .         .     (32) 

and  j*=Z0GothPl  (33) 

Hence  the  Tanh  PI  and  the  Goth  PI  sum  up  mathematically 
the  effect  of  the  reflections  at  the  ends  of  the  cable. 

If  in  the  equation  (32)  we  put  x  =  I  and  therefore  V  =  F2 
we  have 

F2  =  F!  [Cosh  PI  -  Tanh  PI  Sinh  P/]  .     (34) 

or  F^FiSechPJ.        .         .        .        .     (35) 

This  gives  us  an  expression  for  the  potential  difference  of  the 
two  sides  of  the  cable  at  the  far  end  when  a  voltage  V\  is  applied 
at  the  sending  end. 

Again  from  (28)  we  have 

i^TanhPZ  .         .         .     (36) 

^0 

and  the  two  last  expressions  give  us  therefore  the  current  into 
the  cable  at  the  sending  end  and  the  voltage  at  the  distant  end 
when  that  end  is  open. 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  81 

Substituting  this  last  value  (36)  for  /i  in  the  general  equation 
(25)  for  the  current,  we  have  for  the  current  I  at  any  distance  x 
the  expression 

!=-£  [Tanh  PI  Cosh  Px -  Sinh  Px]  (37) 

^o 

If  we  refer  hack  to  equation  (13)  for  the  current  in  an  infinite 
cable  at  any  distance  x  from  the  sending  end  we  see  that  it  can 
be  written 

I=p  [Cosh  Px- Sinh  Px]        .        .        .     (38) 

and  on  comparing  the  last  two  equations  it  will  be  evident  that 
the  effect  of  making  the  cable  finite  in  length  is  to  introduce 
the  quantity  Tanh  PI  in  both  the  formulae  for  the  current  and 
voltage  at  any  point.  Thus  for  the  infinitely  long  cable  the 
equations 

F=  F,  [Cosh  Px  -  Sinh  Px]       .  (39) 

and  I=p  [Cosh  Pa; --Sinh  Pa]       .        .        .     (40) 

give  us  the  voltage  and  current  at  any  distance  x  from  the 
sending  end,  whilst  for  the  finite  cable  of  length  I  we  have 

7-  F!  [Cosh  Px  -  Tanh  PI  Sinh  Px]        .        .     (41) 

and  1=  ~  [Tanh  PI  Cosh  Px  -  Sinh  Px]  .     (42) 

^0 

These  formulae  show  us  that  the  values  for  the  current  and 
voltage  in  an  infinite  cable  become  greatly  modified  when  we  cut 
off  a  length  and  make  it  finite  in  length. 

The  reason  for  this  is,  as  above  stated,  that  when  the  cable  is 
finite  in  length  the  current  and  voltage  at  any  point  are  due  to 
the  superposition  of  an  infinite  number  of  effects  due  to  the 
repeated  reflection  at  the  ends.  We  may  in  fact,  as  Dr.  A.  E. 
Kennelly  has  shown,  derive  the  formulae  for  the  cable  of  finite 
length  by  a  process  of  summation  of  these  direct  and  reflected 
currents.1 

Thus  suppose  a  voltage  V\  is  applied  at  the  sending  end,  this 
travels  up  the  cable  of  length  I  and  at  the  far  end  becomes 
attenuated  to  V\e~1>l. 

1  See  A.  E.  Kennelly,  "  On  the  Process  of  building  up  the  Voltage  and  Current 
in  a  Long  Alternating  Current  Circuit,"  Proc.  of  the  American  Academy  of  Arts  and 
Sciences.  Vol.  XL! I.,  p.  710,  May,  11)07. 

E.C.  G 


82          PROPAGATION   OF   ELECTRIC   CURRENTS 

At  the  open  end  this  potential  difference  is  reflected  and 
doubled  on  reflection  by  the  summation  of  the  arriving  and 
reflected  potentials.  Hence  it  jumps  up  on  reflection  to  2  V\  e~w. 
The  reflected  wave  of  potential  runs  back  attenuating  as  it  goes 
to  FI  e~2Pl,  and  is  reflected  at  the  closed  sending  end  of  the  cable 
with  sign  changed  as  explained  in  Chapter  II.  The  reflected 
wave  again  returns  to  the  receiving  end,  at  which  it  has  attenu- 
ated to  —  FI  e~3p?  and  this  is  doubled  on  reflection  to  2  VI€~BI>I 
and  so  on. 

Hence  at  the  receiving  end  the  actual  potential  difference  is 
the  sum  of  all  these  separate  voltages,  or 

Fa=2.F1(€-«-€-8«-f€-6«-€-^etc.)      .        .    (43) 

The  series  in  the  brackets  is  a  geometrical  progression  with 
ratio  —  €~2Pl  and  hence  we  have, 


_ 
PI 

or  F^FjSechPZ  ....     (44) 

The  hyperbolic  function  Sech  PI  thus  sums  up  the  effect  of 
all  the  repeated  reflections  at  the  ends. 

The  student  will  be  assisted  to  comprehend  the  nature  of  this 
process  by  considering  a  similar  effect  in  the  case  of  light. 
Suppose  a  candle  placed  in  an  otherwise  dark  room.  The 
illumination  at  any  point  would  have  a  certain  value  depending 
on  the  distance  from  the  candle.  If  then  a  mirror  were  placed 
at  this  point,  the  illumination  just  in  front  of  the  mirror  would 
be  equal  to  that  due  to  the  candle  together  with  that  due  to 
another  candle  assumed  to  be  placed  as  far  behind  the  mirror  as 
the  first  candle  is  in  front  of  it,  in  other  words  at  the  position 
of  the  optical  image  of  the  first  candle,  the  mirror  being  then 
supposed  to  be  removed.  Hence  the  single  mirror  produces  on 
the  illumination  the  effect  of  a  second  candle.  In  other  words 
it  doubles  the  illumination.  Imagine  then  that  a  second  mirror 
is  placed  behind  the  candle  so  that  the  candle  stands  between  the 
two  mirrors  ;  the  result  will  be  that  certain  rays  will  be  reflected 
backwards  and  forwards  and  the  illumination  at  a  point  any- 
where between  the  mirrors  will  be  the  same  as  if  the  mirrors 
were  removed  and  an  infinite  number  of  candles  were  placed  in 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  83 

positions  coinciding  with  the  optical  images  of  the  single  candle 
formed  by  repeated  reflections  in  the  mirrors. 

It  will  be  noticed  that  in  all  these  formulae  \ve  are  concerned  with 
the  hyperbolic  functions  of  complex  angles.  Since  the  propaga- 
tion constant  P  and  therefore  the  propagation  length  Px  or  PI 
are  complex  quantities,  viz.,  ax-\-j/3x  or  al+jfil  the  hyperbolic 
functions  are  themselves  vectors,  and  we  must  obtain  their  values 
by  the  rules  given  in  Chapter  I. 

Thus  P  =  "+JP  and  Pl 

and  Cosh  PI  =  Cosh  (« 

=  Cosh  al  Cos  fil+j  Sinn  al  Sin  ftl. 

Since  Sech  PI  =  ^Qstl  pf  we  can  obtain  the  value  of  SechP/ 

by  reciprocating  Cosh  PI  after  its  vector  value  has  been  thrown 
into  the  form  A/6L 

For  example,  suppose  a=Ol,  £=0-1,  Z  =  10. 
Their  PI  =  1  +/  1  =  1-414  /45°. 

Cosh  Pl=  Cosh  (1+J  l)  =  Cosh  1  Cos  1+;  Sinh  1  Sin  1. 
The  1  here  in  Cos  1  and  Sin  1  means  an  angle  of  1  radian  or 
180/77  degrees  =  57°  17'  45".     Hence  from  the  tables 

Cosh  (1  +/  1)  =  1  •  5431  x  0  •  541  +j  1  •  1752  x  0  -841 
=  0-835  +j  0-988  =  1  •  3  /49°  45'. 


Hence  Sech  (1  +j  1)  =  0-77  /49°  45'. 

Again,  if  o  =  0-l,  0  =  0-1,  and  Z  =  20 

Pl  =  2+j  2  =  2-828/45° 
Cosh  (2+y  2)  =  Cosh  2  Cos  2+/  Sinh  2  Sin  2 
=  -  3  •  7622  x  0  •  416  +j  3  •  6269  x  0  •  909 

-1  -565+y  3-  297-3  -66  /115°  24^. 
Hence  Sech  (2  +;  2)  =  0-27  /115°  24'. 

If  a  =  0-1,  p  =  0-3,  1  =  5,  PI  =  1  •  6  /71°35\ 

Cosh  (0-5+y  1-5)  =  Cosh  0  •  5  Cos  1'5+j  Sinh  0-5  Sin  1-5 
=  1-  1276x0- 071 +y  0-521x0  -997 
=  0-080+y  0-520  =  0 -526/81    1-V 
and  Sech  (0  •  5  +;  1  •  5)  =  1  •  9  /$i°~W. 

It  can  be  easily  proved  in  the  same  way  that  if 

1-5-1-5  /84°  17'  Sech  PI  =  6 •  0  /64"  23 '  nearly. 

G  2 


84          PROPAGATION   OF   ELECTRIC   CURRENTS 

It  will  be  seen  therefore  that  for  various  ratios  of  /3/a  and 
values  of  I  V«2+/3'2  the  value  of  the  size  of  Sech  PI  may  be 
greater  than  unity. 

Referring  then  to  the  formula  (35)  for  the  ratio  of  the  voltage 
at  the  open  receiving  end  of  a  cable  to  that  at  the  sending  end, 
viz., 

?2=  Sech  PI 

it  is  clear  that  since  Sech  PI  can  have  a  size  greater  than  unity, 
the  size  of  V^  or  the  numerical  value  of  the  voltage  at  the 
receiving  end  can  be  greater  than  the  numerical  value  of  the 
voltage  at  the  sending  end. 

Thus,  referring  to  the  calculations  just  given,  if  a  =  0*1  and 
/3  =  0'3  and  the  length  of  the  cable  is  five  miles,  then  since 
Sech  PI  in  this  case  is  1'9  /81°  15',  it  follows  that  the  voltage 
across  the  cable  ends  at  the  receiving  end  is  1'9  times  the  voltage 
applied  at  the  sending  end.  In  other  words  there  is  a  considerable 
rise  in  voltage  along  the  cable,  instead  of  a  fall,  entirely  due  to 
the  action  of  reflection  at  the  ends  of  the  cable. 

It  is  of  course  obvious  that  there  will  in  general  be  a  consider- 
able difference  in  phase  between  the  voltages  at  the  sending  and 
receiving  ends,  whilst  the  actual  numerical  value  of  the  voltage 
at  the  open  receiving  end  may  be  less  than,  equal  to,  or  greater 
than  that  at  the  sending  end. 

4.  Propagation  of  Current  along  a  Line  Short 
Circuited  at  the  Receiving  End. — We  have  next  to 
consider  the  case  of  a  line  short  circuited  at  the  receiving  end, 
having  a  simple  periodic  electromotive  force  V\  applied  at  the 
sending  end.  Then  the  voltage  Vz  at  the  receiving  end  is  zero. 
Hence  in  the  general  equations  (23)  and  (25),  viz., 

V=  F!  Cosh  PX-I!  Z0  Sinh  Px  .  .     (45) 

1=1,  Cosh  Px-^  Sinh  Px  .         .     (46) 

^o 

Let  us  put  Vz  =  0,  I  =  /2,  x  =  I,  and  eliminate  Ii,  then  we 
have 

/  (47) 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  85 

This  gives  us  the  current  through  the  short  circuit  at  the 
receiving  end.  Also  from  the  equation  (45),  putting  V  =  0  and 
x  =  I,  we  have 

0=  F!  Cosh  PI  -I,  Z0  Sinh  PL 

Hence  J1  =  p  Goth  PZ      ...  .     (48) 

and  from  (48)  and  (47)  we  have 


.         .     (49) 

or  j=SechJPZ  .....     (50) 

This  gives  us  the  ratio  of  the  current  at  the  receiving  end  to 
that  at  the  sending  end,  and  it  is  clear  that,  since  the  size  of 
Sech  PI  can  be  greater  than  unit}7,  the  current  at  the 
receiving  end  can  be  larger  than  the  current  at  the  sending  end. 

It  is  easy  to  show,  as  in  the  case  of  the  cable  open  at  the  far 
end,  that  this  increase  is  due  to  the  accumulated  effects  of 
reflection  at  the  two  ends. 

The  ratio  Vi/Ii  =  Z\  is  called  the  final  sending  end  impedance, 
and  from  equation  (48)  it  is  seen  that 

l   ....  (51) 


Since  Z0  =     ,       .-.  is  a  vector  quantity  and  since  Tanh  PI 
VS+jpC 

is  a  vector,  it  follows  that  Z\  is  a  vector  quantity,  and  this 
impedance  is  said  to  be  measured  in  vector-ohms,  meaning  that 
the  size  is  measured  in  ohms  but  that  an  angle  giving  direction 
is  appended.  The  ratio  Fi//2  =  ^2  is  called  the  final  rccdruni 
cud  impedance,  and  from  equation  (47)  we  have 

^  =  ^2  =  ^0SinhPZ  .     (52) 

^2 

We  can  measure  experimentally  for  any  line  the  values  of 
FI,  Ii,  and  I2,  and  hence  determine  Z\  and  Z2.  Suppose  the 
ratio  FI//I  is  measured  with  the  far  end  of  the  line  open  or 
insulated.  Let  this  value  of  Z±  be  denoted  by  Zs.  Then  from 
equation  (30)  we  have 


Suppose  FJA  is  then  measured  with  the  far  end  of  the  line 


86          PROPAGATION   OF   ELECTRIC   CURRENTS 

short  circuited,  and  call  the  ratio  under  these  conditions  Zn 
then  from  (48)  we  have 

PI. 


Hence  Zf  Zc  =  Zj,  or 

Z,=  V^ZC  .  •       .         .         .         .     (53) 

Hence  the  initial  sending  end  impedance  is  the  geometric 
mean  of  the  final  sending  end  impedances  with  the  far  end  open 
and  the  far  end  closed.  This  measurement  is  the  hest  means  of 
finding  the  value  of  Vlt-\-jpL/VS+jpC  for  any  actual  line. 

5.  Propagation  of  Simple  Periodic  Currents 
along  a  Transmission  Line  having  a  Receiving 
Instrument  of  known  Impedance  at  the  End. 

This  is  the  practical  telephone  problem  to  the  consideration  of 
which  all  that  has  previously  been  given  is  preliminary.  We 
assume  that  we  have  a  line  of  known  primary  constants  R,  L, 
C,  S,  and  therefore  known  attenuation  constant  a  and  wave 
length  constant  /3,  and  that  a  receiving  instrument  of  known 
impedance  Zr  is  inserted  across  the  line  at  the  receiving  end. 
Assuming  we  apply  a  simple  periodic  electromotive  force  V\  at 
the  sending  end,  the  problem  before  us  is  to  calculate  the 
current  and  voltage  at  the  receiving  end,  or  at  any  distance. 

If  V%  and  /2  are  the  potential  difference  and  current  at  the 
receiving  end,  then  the  impedance  of  the  receiving  instrument  ZT 
is  defined  by  the  relation  V*  =  I<*Zr.  As  V^  and  72  can  be 
measured  by  suitable  methods,  we  can  always  find  Zr. 

Referring  again  to  the  fundamental  equations  (23)  and  (25)  we 
have 

V=  V,  Cosh  Px  -  1,  ZQ  Sinh  Px    .         .         .     (54) 

y 
/=/!  Cosh  Px-^r  Sinh  Px         .         .         .     (55) 

Substituting  for  V,  I,  and  x  the  values  at  the  receiving  end,  viz., 
F2,  /2,  and  /,  we  have 

F2  =  J2  Zr  =  Fj  Cosh  PI  -  1,  ZQ  Sinh  PI         .         .     (56) 

I2  =  I,  Cosh  PI  -  ~  Sinh  PI      .  (57) 


ELECTRIC   CURRENTS   IN   TELEPHONE   CABLES     87 
Eliminating  1\  we  have 


2      ZQ  Sinh  Pl+Zr  Cosh  PI 
and  eliminating  J2  we  have 


T_ViZ0  Gosh  Pl+Zr  Sinh  PZ 
l~Z*Zr  Cosh  P/  +^o  Sinh  P£    ' 


These  expressions  give  us  the  current  at  the  receiving  and 
sending  ends  respectively.  Hence  also 

—  Cosh  PI  +^  Sinh  PI  (60) 

•H  ^0 

On  comparing  the  above  formula  with  the  corresponding 
formula  (49)  for  the  cable  short  circuited  at  the  receiving  end, 
we  see  that  the  effect  of  the  receiving  instrument  is  to  add  a 

r? 

term  -nr  Sinh    PI,    and   so  make  the  ratio  Ii/Iz  larger.     It  is 

**o 

possible,  however,  for  1%  to  be  greater  than  /i.  From  the  above 
formulae  (59)  and  (58)  we  can  obtain  expressions  for  the  final 
sending  end  impedance  Z\  =  FI//I  and  for  the  final  receiving 
end  impedance  Z%  =  V\II%,  viz., 

F!_      Zr  Cosh  Pl+ZQ  Sinh  PI 
*i-Ii  -  *o  ZQ  Cosh  Pi  +  zr  Sinh  PI 

l      .        •        •     (62) 


The  above  expressions  can  be  simplified  by  taking  advantage 
of  two  well-known  theorems  in  circular  and  hyperbolic 
trigonometry. 

Theorem  I.     If  6  is  any  circular  angle  such  that  tan  6  =  —  , 
and  if  $  is  any  other  angle,  then 


A  Sin  $+B  Cos  <£=  </A*+B*  Sin 
If  B/A  =  tan  0,  then 

^        =  Sin  0  and 


but  Sin  ((/>  +  0)  =  Sin  <f>  Cos  0  +  Cos  <f>  Sin  0.      Hence,  substi- 
tuting the  values  of  Sin  6  and  Cos  0,  we  have 

A  Sin  0+J3  Cos  <£=  x/^r+5-  Sin  (^  +  0)    .         ,     (63) 


88          PROPAGATION   OF   ELECTRIC  CURRENTS 
Theorem  II.     If  y  is  any  hyperbolic  angle  such  that 

73 

tanh  y  =  -T,  and  if  5  is  any  other  hyperbolic  angle,  then 

A. 


A  Sinh  S  +  £  Cosh  8=  JA*-&  Sinh  (8+y), 
Sinh        B 


For 
and  Cosh  2y  -  Sinh  2y  =  1  . 

Hence        Sinh  y  =  and  Cosh  y  = 


But  Sinh  (8  +  y)  =  Sinh  8  Cosh  y+Cosh  8  Sinh  y 

Hence  A  Sinh  8+5  Cosh  8=  VA2-B*  Sinh  (8+y)  .        .     (64) 

Again,  from  the  fundamental  equation  (23) 

F2=  V,  Cosh  PZ-Ij  ZQ  Sinh  PZ      .         .         .     (65) 
and  from  the  value  obtained  for  I\  in  (59)  we  have 
T   y      T/^o  Cosh  Pl+Zr  Sinh  PI 
ll    -°r    %  Cosh  Pl  +  Z,  Sinh  PI 


Hence,  substituting  (66)  in  (65),  we  have 


T/      T7  3  n     K'  P7      ^o  Cosn  r  ,p  , 

F2=  7,  1  Cosh  PZ  -  UTCosh  Pl  +  Z,  Sinh  PzSmh   P/'          (6?) 

or  since  Cosh  2PZ  —  Sinh  2P^  =  1  we  have 

V    7, 

TT  _  '  1  ^J  r 

*~ 


Q  Sinh  Pl+Zr  Cosh  PI 
Accordingly  by  the  aid  of   the   Theorem  II.  we  can    write  the 
formulae  for  the  currents  and  final  impedances  as  follows  :— 

^=-^^2  Cosech  (Pl+y)     •  •     (69) 

T7 

I^^Coth  (P^+y)  .  .     (70) 

^2=  JZ^-Zf  Sinh  (P/+y)      .          .         .     (71) 
^!  =  ^0Tanh(PZ+y)   .         .         .         .     (72) 

where  Tanh  y=^  or  y-  Tanh-1  (^)  (73) 

^o  \^o/ 

Hence  it  follows  that 

I^l!  Cosh  y  Sech  (P2  +  y)  .  .  .      (74) 

rf 

Also  from  (68),  bearing  in  mind  that  Tanh  y  =  ~  and  therefore 

Z 

Sinh  y  =     ,       r    =,  we  can  express  the  ratio  TV  FI  by 

v^o  —  ^*- 

72=F!  Sinh  y  Cosech  (P/  +  y)       .         .         .     (75) 


ELECTRIC  CURRENTS  IN  TELEPHONE  CABLES  89 

A  consideration  of  these  last  five  formulae  and  comparison  of 
them  with  the  similar  formulas  for  the  short  circuited  cable 
shows  that  the  introduction  of  the  receiving  instrument  of  im- 
pedance Zr  has  the  same  effect  as  if  the  line  were  made  longer 
by  an  amount  I'  such  that  PV  =  y  and  was  then  short 
circuited  at  the  receiving  end.  At  the  same  time  the  effect 
of  this  lengthening  is  to  cause  an  alteration  in  the  effective 
initial  sending  end  impedance  as  far  as  the  current  at  the 
receiving  end  is  concerned,  but  not  for  the  sending  end  current. 

We  have  shown  (equation  (52))  that  the  final  receiving 
end  impedance  Vi/f*  in  the  case  of  a  line  short  circuited  at  the 
receiving  end  is  Z2  =  ZQ  Sinh  PL 

And  also  that  the  same  quantity  for  the  line  with  receiving 
instrument  of  impedance  Zr  at  the  end  is  (by  equation  (62)  ) 

given  by 

^2  =  ^0  Sinh  Pl+Zr  Cosh  PL 

Hence  if  we  denote  the  final  receiving  end  impedance  of  the 
short  circuited  line  by  Z^  we  have 

Z      .  (7G) 


When  the  line  is  very  long  Coth  PI  approximates  to  unity  and 
then 


CHAPTER  IV 

TELEPHONY  AND  TELEPHONIC  CABLES 

1.  The  Principles  of  Telephony. — Telephony  is  the 
art  and  science  of  transmitting  articulate  speech  by  means  of 
electric  currents  between  two  places  connected  by  a  wire  or  cable. 
The  conductor  may  be  either  a  pair  of  overhead  wires  or  a  single 
wire  with  earth  return,  or  a  twin  cable. 

At  one  end  of  this  conductor  is  placed  a  telephone  transmitter, 
which  comprises,  generally  speaking,  an  induction  coil,  the 
secondary  circuit  of  which  is  connected  to  the  pair  of  line 
wires  or  to  the  line  wire  and  the  earth.  In  the  primary  circuit 
of  the  coil  is  included  a  battery  and  a  microphone.  This  last 
consists  in  one  form  of  a  shallow  circular  metal  box  with  a 
solid  back  ;  closed  in  front  by  a  diaphragm  of  flexible  metal 
which  is  insulated  by  a  ring  of  ebonite  from  the  box  itself. 

The  cavity  is  filled  with  granulated  graphitic  carbon.  Wires 
are  connected  to  the  diaphragm  and  to  the  box. 

An  electric  circuit  is  thus  formed,  of  which  the  granulated 
carbon  is  part. 

This  arrangement  constitutes  the  microphone,  and  it  is  joined 
in  series  with  the  battery  and  with  the  primary  circuit  of  the 
induction  coil.  If  the  carbon  granules  are  compressed  by 
pressing  in  the  diaphragm  the  resistance  of  the  circuit  is 
reduced  and  more  current  flows  through  the  primary  circuit  of 
the  coil  and  hence  induces  a  current  in  the  secondary  circuit, 
which  flows  through  the  line. 

If  articulate  speech  is  made  in  front  of  the  diaphragm  the 
rapid  changes  of  air  pressure  which  constitute  sound  cause 
a  corresponding  movement  of  the  diaphragm  and  therefore 
equivalent  changes  in  resistance  in  the  carbon  granules.  Hence 
a  secondary  current  is  sent  into  the  line  the  variations  in  which 


TELEPHONY  AND  TELEPHONIC  CABLES    91 

more  or  less  perfectly  follow  the  changes  of  air  pressure  in  front 
of  the  diaphragm. 

The  motion  of  the  air  molecules  when  transmitting  a  sound 
wave  is  to  and  fro  in  the  direction  of  transmission,  but  the 
amplitude  of  their  acoustic  motion  is  extremely  small. 

Lord  Eayleigh  determined  the  amplitude  of  this  air  motion 
for  the  sound  of  a  whistle  giving  a  note  having  a  frequency  of 
2730,  which  was  loud  enough  to  be  heard  at  a  distance  of  820 
metres  in  every  direction.1  This  amplitude  he  found  to  be 
0*081  of  one  millionth  of  a  centimetre  or  0*00081  ^  where  ^  is 
the  thousandth  part  of  a  millimetre.  This  is  about  one 
thousandth  part  of  the  wave  length  of  a  ray  of  red  light  and 
shows  how  extremely  small  an  air  motion  the  normal  human  ear 
is  capable  of  appreciating.  In  the  case  of  articulate  sounds  this 
motion  of  the  air  particles  is  a  highly  irregular  one,  but  in  the 
case  of  musical  sounds  or  prolonged  vowel  sounds  the  motion  is 
a  regularly  repeated  or  cyclical  one  which  is  to  and  fro  in  the 
line  of  propagation  of  the  sound.  We  can  graphically  represent 
it  by  the  displacement  of  a  point  which  moves  uniformly  along 
a  straight  line  and  at  the  same  time  executes  a  vibratory  motion 
at  right  angles  to  that  line  which  copies  the  to  and  fro  motion  of 
the  air  particle  in  the  line  of  propagation.  We  then  obtain  for 
continuous  sounds  a  wavy  line  which  is  called  the  graph  or  wave 
form  of  the  sound. 

The  curves  in  Fig.  1  represent  the  wave  forms  of  five  vowel 
sounds,  A,  E,  1, 0,  U,  pronounced  in  the  Continental  manner.  If 
the  sound  recorded  is  that  of  a  tuning  fork  or  open  organ  pipe 
gently  blown  the  wave  form  is  a  simple  periodic  curve  such 
that  the  displacement  or  ordinate  y  at  any  time  t  is  given  by 
the  expression  y  =  Y  Sin  pi  where  Y  is  the  maximum  ordinate 
and  p  =  2-7T  times  the  frequency  n. 

On  the  other  hand,  if  the  sound  is  a  consonantal  sound  or 
noise,  the  wave  form  is  an  irregular  non-repeated  curve.  If  it 
is  a  periodic  or  repeated  curve  the  maximum  amplitude  is 
determined  by  the  loudness  of  the  sound  and  its  wave  length  or 
period  by  the  pitch. 

1  Sec  Lord  Rayleigh,  Proc.  Roy.  Soc.,  Vol.  XXV I.,  p.  248,  1877,  or  Collected  I><IJHTS. 
Vol.  I.,  p.  328. 


92         PROPAGATION   OF   ELECTRIC   CURRENTS 

to  ^  ^ 


TELEPHONY  AND  TELEPHONIC  CABLES    93 

When  any  sound  or  speech  is  made  in  proximity  to  the 
diaphragm  of  the  microphone  the  aerial  vibrations  create  a  more 
or  less  similar  motion  of  the  diaphragm,  and  a  variation  in  the 
resistance  of  the  carbon  granules  takes  place,  which  in  turn 
causes  the  current  into  the  line  wire  to  be  varied  in  a  manner 
somewhat  similar  to  the  variations  of  the  air  pressure  and 
motion  in  front  of  the  diaphragm. 

Owing  to  the  fact  that  the  diaphragm  has  a  natural  period 
of  vibration  of  its  own,  this  current  variation  in  the  line  is 
not  an  exact  copy  of  the  air  pressure  variation,  but  it  is 
sufficiently  like  it  to  achieve  practical  telephony.  We  may  then 
assume  that  there  is  in  the  line  wire  a  current  the  wave 
form  of  which  at  the  sending  end  is  somewhat  similar  to  that 
of  the  wave  form  of  the  air  motion. 

This  current  flows  along  the  line,  and  being  a  periodic  current 
it  is  attenuated  as  it  flows.  At  the  receiving  end  it  enters 
the  telephone  receiver,  which  is  generally  a  Bell  magneto 
telephone  consisting  of  a  permanent  magnet  of  bar  or  horse- 
shoe form,  round  the  poles  of  which  are  coils  of  wire 
inserted  in  series  with  the  line  wire.  In  close  proximity  to  the 
poles  of  this  magnet  is  a  flexible  diaphragm  of  iron  (ferrotype 
plate).  When  the  periodic  current  from  the  line  flows  through 
the  coils  wound  on  the  magnet  it  slightly  increases  or  decreases 
the  magnetism  and  attracts  more  or  less  the  iron  diaphragm. 
The  result  is  that  the  diaphragm  of  the  receiving  instrument 
experiences  vibrations  which  are  approximately  a  copy  of  the 
variations  of  the  line  current  and  therefore  of  the  vibrations  of 
the  diaphragm  of  the  transmitter.  The  air  in  proximity  to  the 
receiving  diaphragm  is  therefore  set  in  vibration  in  a  manner 
which  is  not  very  dissimilar  to  that  of  the  diaphragm  of  the 
transmitter  and  therefore  to  that  of  the  air  in  proximity  to  the 
latter.  We  thus  repeat  in  a  distant  place  sounds  made  near  the 
transmitter.  This  repetition  is,  however,  far  from  being  perfect. 
The  transmission  of  articulate  sounds  is  wonderfully  assisted  by 
the  power  of  the  human  intelligence  to  guess  from  a  very 
imperfect  repetition  the  significance  of  the  sound  as  a  word 
spoken  to  the  transmitting  diaphragm.  There  is,  however,  a 
limit  to  this  guesswork,  and  beyond  a  certain  point  a  sound  may 


94         PROPAGATION   OF   ELECTRIC   CURRENTS 

be  heard  but  it  has  no  meaning.  This  constitutes  the  limitation 
of  telephony,  and  we  have  in  the  next  place  to  consider  the 
causes  of  this  limitation. 

2.  Fourier's  Theorem. — If  we  have  any  single  valued 
periodic  curve,  that  is  one  having  only  one  value  of  the  ordinate 
to  one  value  of  the  abscissa  and  repeating  itself  at  regular 
intervals,  then,  no  matter  how  irregular  the  curve  may  be  provided 
it  does  not  exhibit  discontinuities,  it  is  always  possible  to  imitate 
this  curve  exactly  by  adding  the  ordinates  of  superimposed 
simple  periodic  or  sine  curves  of  suitable  amplitude  and  phase 
difference,  having  wave  lengths  which  are  in  integei'  relation  to 
each  other.  Thus,  for  instance,  if  we  draw  sine  curves  having 
wave  lengths  in  the  ratio  of  1,  J,  J,  J,  etc.,  we  can  cut  thin 
sheets  of  zinc  so  that  these  curves  form  the  outline  of  one  edge 
(see  Fig.  2),  and  these  templates  can  be  used  to  draw  sine  curves 
of  certain  relative  amplitude  and  phase  difference  relatively  to 
one  another. 

We  can  then  add  together  the  ordinates  of  the  several  com- 
ponents corresponding  to  any  one  abscissa,  to  form  the  ordinate 
of  a  new  compound  curve  which  is  then  said  to  be  made  by  the 
synthesis  of  these  several  sine  curves.  There  is  no  difficulty  in 
carrying  out  this  synthetic  process. 

It  is  rather  more  difficult  to  perform  the  inverse  process,  i.e., 
when  given  an  irregular  but  periodic  single  valued  curve  to  find 
the  sine  components  of  which  it  is  built  up,  but  it  can  be  done  in 
virtue  of  Fourier's  Theorem,  which  is  as  follows : — 

Let  y  be  the  ordinate  of  any  periodic  curve  which  is  single 
valued  and  without  discontinuities ;  then  y  can  be  expressed  by 
the  series 

y=A^Al  Smpt  +  Bi  Goapt+.-*z  Sin  Zpt+B^  Cos  Zpt 

+  A3  Sin  3pt  +  Ba  Cos  3pt  etc.      .         .  (1) 

where  p=  ^\T  and  T  is  the  periodic  time  of  the  fundamental 
sine  curve  assuming  it  to  be  described  by  a  point  which  moves 
with  uniform  velocity  in  a  horizontal  direction. 

Accordingly  pt  is  the  abscissa  corresponding  to  y,  such 
abscissa  being  measured  from  the  zero  point  of  the  fundamental 
sine  curve. 


TELEPHONY  AND  TELEPHONIC  CABLES    95 


FIG.  2. — Templates  of  Curves  representing  Harmonic 
Sine  Curves. 


96         PROPAGATION   OF   ELECTRIC   CURRENTS 

The  problem  then  is  to  find  the  value  of  the  constants 
Aot  A^  Bit  AZ,  B%,  etc.  This  can  be  done  by  the  aid  of  the 
following  theorems  :— 

1.  If  9  is  any  angle,  then  the  average  value  of  Sin  0  or  Cos  0 
taken  at  equal  small  angular  distances  over  four  right  angles 
is  zero. 

2.  The  average  value  of  both  Sin2  9  and  Cos2  9  taken  in  the 

.    1 
same  way  is  ^. 

3.  The  average  value  of  Sin  9  Cos  0  and  of  Sin  n  9  Sin  m  9 
or  Cos  n  9  Cos  m  9  taken  in  the  same  manner  over  four  right 
angles  is  zero. 

The  truth  of  Theorem  1  is  obvious.  If  we  describe  a  sine 
curve  extending  over  a  complete  period,  taking  this  as  360°,  and 
draw  equi-spaced  ordinates,  then  it  is  clear  that  for  every  positive 
ordinate  there  will  be  an  equal  negative  ordinate  separated  by 
abscissa  equal  to  180°,  and  when  we  add  their  values  algebraically 
the  sum  of  the  whole  number  is  zero,  and  therefore  the  average 
value  is  zero. 

Theorem  1  expressed  in  the  language  of  the  integral  calculus  is 
otherwise  proved  as  follows  :  — 

f2» 

3in  0  d  0  .         .         .         .     (2) 


where  M  Sin  6  stands  for  the  average  or  mean  value  of  Sin  9 
taken  at  equal  angular  intervals  cW  between  0  and  STT.     If  then 

27T 

0  —  pt  —  -r^t,  then 

1  CT 

iR\    Bmptdt    ....     (3) 

J-  Jo 


l  +  l)  =  0. 

Theorem  2  can  be  proved  as  follows  :— 

Since     Cos  20  =  Cos  -0  -  Sin  20  =  1  -  2  Sin  20  =  2  Cos  -0  -  1 

we  have  Sin2  0  =  ^  -  g  Cos  20 

and  Cos'2  0  =    +    Cos  20. 


TELEPHONY  AND  TELEPHONIC  CABLES    97 

The  average  value  of  Sin  2#  must  therefore  be  -^  because  the 

average  value  of  Cos  20  taken  at  small  equal  angular  intervals 
between  6  =  0  and  0  =  2vr  is  zero.     The  same  for  Cos  2#. 
Theorem  3  is  also  easily  proved  ;  for 

Sin  (n+m)  0  =  Sin  nO  Cos  ??i0-|-Cos  nO  Sin  mO 
Sin  (n  —  m)  0  =  S'm  n&  Cos  mO  —  Cos  nO  Sin  mO. 

Hence    Sin  nO  Cos  m6  =  -^  Sin  (n+m)  0+^  Sin  (n  —  m)  0. 

Accordingly  whatever  n  and  m  may  be,  the  average  value  of 
Sin  n  6  Cos  m  6  must  be  zero  because  the  average  values  of 
Sin  (tt+Hi)  0  and  Sin  (n—  m)  0  are  individually  zero. 

Again  it  can  be  proved  in  the  same  way  that 

Sin  nO  Sin  mO  =  ^  Cos  (n  —  m)0  —  -^  Cos  (n-\-m)0 

Cos  nO  Cos  mO  =  n  Cos  (w  — w)0-fg  Cos  (?i+w)0. 

It  is  clear  then  that  the  average  values  of  Sin  n  6  Sin  in  0  and 
Cos  n  6  Cos  in  6  are  zero  except  when  n  =  m,  in  which  case  their 

average  values  are  g. 

Eeturning  then  to  the  expression  first  given  by  Fourier  for  the 
ordinate  y  of  a  single  valued  continuous  curve  by  the  series 

y  =  A0+Al  Smpt+Bi  Cospt  +  Az  Sin  2pt+B2  Cos  2  pt  etc., 
we  have  to  show  how  the  constants  in  this  series  can  be  found. 

Suppose  a  number  of  equi-spaced  ordinates  yit  y%,  y3  to  be 
drawn  to  the  curve  over  one  complete  period.  Then  the  average 
value  of  these  ordinates  throughout  this  period  is  the  value  of  AQ 
because  the  average  value  of  all  the  Sine  and  Cosine  terms  is 
zero. 

Again  let  us  multiply  both  sides  by  Sinp£  and  take  the  average 
value  throughout  the  period,  we  have 

y  Sin  pt  =  A0  Bmpt+A^  8wPpt-\-Bi  Cos  2^  Sin_p£,  +  etc. 

When  we  take  the  average,  the  value  of  all  the  terms  on  the  right- 

^ 
hand  side  is  zero,  except  A\  Sin  *pt  which  is  equal  to  -<p 

Hence  we  have  the  average  value  of  y  Sin_p£  =  -^  or 

AI  =  twice  the  average  value  of  y  Sin  pt  through  a  period. 
B.C.  H 


98          PROPAGATION   OF   ELECTRIC    CURRENTS 

In  the  same  way  by  multiplying  successively  by  Cos  pt,  Sin  2  pt, 
Cos  2  pt,  etc.,  we  can  prove  that 

BI  =  twice  the  average  value  of  y  Cos  pt, 

A%  =  twice  the  average  value  of  y  Sin  2  pt, 

B2  =  twice  the  average  value  of  y  Cos  2  p£,  etc. 

Accordingly  we  have  the  following  rule  for  analysing  a  com- 
pound periodic  curve  into  its  constituent  harmonics. 

Rule  up,  say,  24  ordinates  at  equal  distances  throughout  one 
complete  period  of  the  curve  and  measure  off  their  lengths. 
Call  them  yif  ?/2,  y3,  2A,  etc. 

Then,  since  360/24  =  15  we  must  look  out  in  the  Tables  the 
values  of  Sin  0°,  Sin  15°,  Sin  30°,  Sin  45°,  Sin  60°,  etc.,  and 
make  a  table  in  columns  as  follows  :— 

Column  I.  has  the  24  numerical  values,  y\y  y%  .  .  .  .  3/24  written 
down  one  above  the  other. 

Column  II.  has  the  values  ?/i  Sin  15,  y%  Sin  30,  y3  Sin  45,  etc., 
written  above  one  another. 

Column  III.  has  the  values  of  y\  Cos  15,  7/2  Cos  30,  y3  Cos  45, 
etc.,  written  one  above  the  other. 

Column  IV.  has  the  values  y\  Sin  30,  y%  Sin  60,  y3  Sin  90,  etc., 
written  one  above  the  other. 

Column  V.  has  the  values  y\  Cos  30,  ?/2  Cos  60,  y3  Cos  90,  etc., 
written  one  above  the  other. 

Column  VI.  has  the  values  y\  Sin  45,  y2  Sin  90,  ?/3  Sin  135, 
and  so  on,  regard  being  taken  to  the  algebraic  sign  of  the  Sine 
or  Cosine. 

We  have  already  shown  (see  Chap.  III.,  §  5)  that 


A  Sin  </>+£  Cos  <£=  \/A*  +  B*  Sin  (<£  +  0)       .         .     (4) 

-pt 
where   tan   6  =   -^  ;    hence   we   can   write   Fourier's   theorem 

in  the  form, 


y=A0+  yA2  +  #i2  Sin  Qrf  +  004-  vW+^22  Sin(2j^  +  02)  etc.   .     (5) 


In  this  case  the  quantities  \/Ai*+B£  *JA£+B?t  etc.,  are  called 
the  amplitudes  of  the  different  harmonics,  and  the  angles  0i,  02, 
etc.,  are  called  the  phase  angles. 

If  the  curve  is  a  periodic  curve  of  such  kind  that  for  every 
ordinate  of  a  certain  length  there  is  another  ordinate  half  a  wave 
length  further  on  of  equal  length  but  opposite  sign,  then  the  first 


TELEPHONY  AND  TELEPHONIC  CABLES 


99 


or  constant  term  AQ  is  zero,  because  the  average  value  of  all  the 
equi-spaced  ordinates  is  then  zero. 

As  an  example  of  the  Fourier  analysis  of  a  complex  periodic 
curve  we  may  take  the  following  l : — 

The  firm  line  curve  in  Fig.  3  is  a  curve  formed  by  adding 
together  the  ordinates  of  three  simple  periodic  or  (dotted)  sine 


FIG.  3.  —  Fourier  Analysis  of  a  Periodic  Curve. 

curves  of  which  the  wave  lengths  are  in  the  ratio  of  1  :  J  :  -J-  and  of 
which  the  amplitudes  are  respectively  4,  2*8,  and  1'6.  These  curves 
are  shifted  relatively  to  one  another  so  that  the  second  harmonic 
lies  15°  behind  the  first  and  the  third  about  4°  30'  behind  the 
first  harmonic.  These  harmonics  are  represented  by  the  three 
dotted  line  curves  in  Fig.  3. 

Hence  the  equation  to  the  firm  line  curve  is 

7/  =  4  Sin  <£  +  2-8  Sin  3  (<£  +  15°)-l-6  Sin  5  (9  +  *°  30')  .     (6) 


1  The  method  of  numerical  calculation  here  given  was  originally  described  by 
Professor  J.  Perry  in  The  Electrician,  Vol.  XXVI1L,  p.  362,  1892. 

H   2 


100       PEOPAGATION   OF   ELECTEIC   CURRENTS 


If  we  shift  the  origin  to  the  zero  point  of  the  principal  sine 
curve,  this  is  equivalent  to  substituting^  —  15°  for  <p  in  the  above 
equation,  and  the  expression  then  becomes, 

y  =  4  Sin  Qrt-15°)  +  2-8  Sin  3^-1*6  Sin  (5  jp*-52°  30')     .     (7) 

We  then  take  the  curve  as  drawn  and  rule  up  12  equi- spaced 
ordinates  at  intervals  of  15°  and  find  by  actual  measurement 
that  these  ordinates  have  the  values  0,  1*5,  2*4,  3'8,  4*0,  2*3, 
-O'l,  0-4,  4'2,  7-0,  6'2,  2'7  and  0. 

We  then  proceed  to  make  two  tables  as  follows : — Table  I. 
contains  the  values  of  Sin  pt,  Cos  pt,  Sin  3pt,  Cos  Qpt,  Sin  5pt, 
Cos  5pt  for  values  of  pt  from  0°  to  180°. 

TABLE  I. 


pt. 

Sin  pt. 

Cos  pt. 

8j* 

Siu  3  pt. 

Cos  3  pt. 

bpt. 

Sin  5  pt. 

Cos  5  pt. 

0 

0 

1-000 

0 

0 

1-000 

0 

0 

1-000 

15 

•259 

•966 

45 

•707 

•707 

75 

•966 

•259 

30 

•500 

•866 

90 

1-000 

0 

150 

•500 

-•866 

45 

•707 

•707 

135 

•707 

-•707 

225 

-•707 

-•707 

60 

•866 

•500 

180 

0 

-1-000 

300 

-•866 

•500 

75 

•966 

•259 

225 

-•707 

-  -707 

15 

•259 

•966 

90 

1-000 

0 

270 

-1-000 

0 

90 

1-000 

0 

105 

•966 

-•259 

315 

-•707 

•707 

165 

•259 

-•966 

120 

•866 

-  -500 

360 

0 

1-000 

240 

-•866 

-•500 

135 

•707 

-  -707 

45 

•707 

•707 

315 

-•707 

•707 

150 

•500 

-•866 

90 

1-000 

0 

30 

•500 

•866 

165 

•259 

-•966 

135 

•707 

-•707 

105 

•966 

-•259 

180 

0 

-1-000 

180 

0 

-1-000 

180 

0 

-1-000 

In  Table  II.  ,  Column  II.,  are  tabulated  the  measured  values  of 
the  12  ordinates  y  of  the  firm  line  curve  taken  at  equi-spaced 
distances  over  the  half  wave  length  represented  by  180°.  In 
Columns  III.  to  VIII.  are  tabulated  the  values  of  y  Sin  pt, 
y  Cos  pt,  y  Sin  3  pt,  y  Cos  3  pt,  y  Sin  5  pt,  y  Cos  5  pt,  and  at  the 
foot  of  each  column  is  given  the  mean  value  of  each  series  of 
numbers;  also  twice  the  mean  values,  which  are  as  shown  above, 
are  the  values  of  the  Constants  A\,  BI,  A3,  B3,  A5,  B5  respec- 
tively. From  this  are  calculated  the  values  of  the  amplitudes 


52,  and  the  phase  angle  tangents 
Bi/Alt  B3/A3,  and  B5/A5. 

Hence  we  can  find  the  phase  angles  themselves  and  arrive  at 
an  expression  for  the  ordinate  of  the  dotted  curve  expressed  as 


TELEPHONY  AND   TELEPHONIC!  ^ 


101 


a  Fourier  series.     On  comparing  the  expression   thus  obtained 
by  calculation,  viz., 

7/  =  3-92  Sin  (^-15°  50')  +  2  •  9  Sin  (3_p  +  0°50') 

-1-55  Sin  (5  pt  -  51°  30')  .         .        .     (8) 

\yith  the  expression  from  which  the  curve  was  drawn  as  given 
under  Fig.  3,  viz., 

7/  =  4-OSin  (^-15°)  +  2-8  Sin  3^-1 -6  Sin  (5jp*-52°  30')  .  (9) 
it  will  be  seen  that  there  are  small  differences  in  the  amplitudes 
and  phase  angles,  but  that  the  calculated  value  of  the  expression 
agrees  substantially  with  the  expression  from  which  the  firm  line 
curve  in  Fig.  3  was  drawn.  The  differences,  such  as  they  are, 
are  due  to  the  fact  that  we  .have  only  measured  12  ordinates 
in  the  half  wave,  but  it  would  require  a  larger  number  to  secure 


a  better  agreement. 


TABLE  II. 


L 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

pt. 

y- 

y  Sin  pt. 

y  Cos  pt. 

y  Sin  3  pt. 

y  Cos  3  pt. 

y  Sin  Upt. 

y  Cos  5  pt. 

0° 

0 

0 

0 

0 

0 

0 

0 

15° 

1*6 

0-388 

1-449 

1-060 

1-060 

1-449 

0-388 

30° 

2-4 

1-200 

1-838 

2-400 

o-ooo 

1-200 

2-078 

45° 

3-8 

2-686 

2-686 

2-687 

-2-687 

-2-687 

-2-687 

60° 

4-0 

3-464 

2-000 

o-ooo 

-4-000 

-3-461 

2-000 

75° 

2-3 

2-222 

0-586 

-1-626 

-1-626 

0-596 

2-222 

90° 

-o-i 

-o-ioo 

o-ooo 

0'1<0 

o-ooo 

-o-ioo 

o-ooo 

105° 

0-4 

0-386 

-0-104 

-0-283 

0-283 

0-104 

-0-386 

120° 

4-2 

3-637 

-2-100 

o-ooo 

4-200 

-  3-637 

-2-100 

135° 

7-0 

4-919 

-4-949 

4-949 

4-949 

-4-949 

4-949 

150° 

6-2 

3-100 

-5-369 

6-200 

o-ooo 

3-100 

5-369 

165° 

2-7 

0-699 

-  2-608 

1-908 

-1-908 

2-608 

-0-706 

180° 

0 

o-oo  j 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

o-ooo 

I 

+  22-731 

+  8-559 

+  19-304 

+  10-492 

+  9-057 

+  15-228 

'  1 

-0-100 

-15-030 

-  1-909 

-10-221 

-14-837 

-7-957 

Net  totals      . 

+  22-631 

-6-471 

+  17-395 

+  0-271 

-5-780  ' 

+  7-271 

Mean  values.      +1-886 

-0-539 

+  1-450 

+  0-022 

-0-482 

+0-606 

Twice  mean 

+3-77 

-1-08 

+  2-9 

+  0-044 

-0-964 

+  1-212 

values 

=*A\ 

=*B\ 

=AS 

=  -#3 

=A$ 

=  B5 

Therefore 
And    tan-i1 


tan-i(--283)    tan-'  ~>  =tan~i  (-015)   tan~i  ^  =tan-i  (-1-257) 


Hence  we  have 
and  =  3-92  Sin 


=  9i  =  -  1  5°  50' 


-  15° 


2'9  Sin 


03  =  0°  50'  =  05  =  -  5  1°  30' 

')  -1-55  Sin  (5  ^-51°  30'). 


102        REOPATO^:I?   ELECTRIC   CURRENTS 

3.   The   Analysis    and    Synthesis    of  Sounds. 

The  analysis  of  a  periodic  curve  into  its  constituent  sine  curves 
in  accordance  with  Fourier's  theorem  is  not  merely  a  mathe- 
matical conception  or  process,  but  it  is  in  accordance  with 
the  facts  of  acoustics. 

We  can  by  certain  appliances  cause  the  oscillatory  motions 
of  sounding  bodies  to  record  the  nature  of  their  vibrations  in 
graphical  form.  Thus  if  we  attach  to  the  prong  of  a  steel 
tuning  fork  a  bristle  and  hold  the  vibrating  fork  near  a  rapidly 
revolving  drum  covered  with  smoked  paper  we  can  make  the 
bristle  record  the  wave  form  of  the  vibration  upon  the  paper. 
It  is  found  that  this  record  is  a  sine  curve.  The  aerial  vibrations 
produced  by  the  fork  and  also  those  produced  by  open  organ  pipes 
gently  blown  are  in  like  manner  simple  sine  vibrations.  Such 
sounds  are  smooth  and  not  unpleasant  to  the  ear,  but  they  are 
wanting  in  character  or  brilliancy.  If,  however,  a  special  sound 
such  as  a  continuous  vowel  sound  is  made,  we  find  by  experiments 
with  the  oscillograph  or  phonograph  that  the  wave  form  is  very 
irregular  although  periodic.  Von  Helmholtz  was  led  by  these 
considerations  to  his  classical  experiment  of  the  synthesis  of 
vowel  sounds.  He  provided  a  number  of  tuning  forks  the 
frequencies  of  which  were  in  the  ratio  1  :  J  :  J  :  J,  etc.,  and  each 
tuning  fork  had  a  hollow  brass  sphere  in  proximity  to  it,  the 
said  sphere  having  an  opening  in  it.  These  spheres  are  called 
resonators,  and  when  constructed  of  such  size  that  the  corre- 
sponding tuning  fork  can  set  the  air  in  it  in  vibration  they 
re-enforce  the  sound,  provided  the  aperture  of  the  resonator  is 
open.  The  tuning  forks  were  maintained  in  vibration  continuously 
by  electromagnets,  and  by  means  of  keys  the  operator  could 
more  or  less  open  the  aperture  of  any  resonator  and  so  mix 
together  sounds  of  harmonic  frequencies  in  various  proportions 
as  regards  amplitude  or  loudness.  Von  Helmholtz  found  that 
he  was  thus  able  to  imitate  various  vowel  sounds,  and  that  these 
latter  are  therefore  compounded  of  various  simple  sine  vibrations 
of  different  amplitude.  The  question  then  arises,  has  the  relative 
difference  of  phase  of  the  simple  sine  components  anything  to 
do  with  the  production  of  the  quality  of  the  sound  ? 

We  know  from  Fourier's  theorem  that  the  wave  form  of  the 


TELEPHONY  AND  TELEPHONIC  CABLES   103 

complex  curve  depends  not  only  on  the  amplitudes  but  on  the 
relative  phase  of  the  component  sine  curves.  The  question  then 
arises  whether  the  ear  when  impressed  by  a  complex  vibration 
takes  note  of  the  difference  of  phase  as  well  as  the  difference  in 
amplitude  of  the  component  harmonics. 

Von  Helmholtz  drew  the  conclusion  from  his  experiments  that 
the  quality  of  the  sound  depended  only  on  the  amplitudes  of  the 
harmonics  and  not  on  their  relative  phase  (see  Helmholtz's 
book  "  Sensations  of  Tone,"  English  translation  by  Ellis, 
Chap.  VI.,  p.  126). 

Helmholtz's  conclusion  is  not  generally  accepted.  Lord 
Rayleigh  (see  "  Theory  of  Sound,"  Vol.  II.,  Chap.  XXIII.)  has 
given  arguments  to  prove  that  the  difference  of  phase  is  not 
without  effect.  Also  Konig,  another  great  acoustician,  asserts 
that  whilst  quality  in  sound  is  mainly  dependent  upon  the  relative 
amplitude  of  the  harmonics  the  difference  of  phase  makes  some 
contribution  to  it. 

Hence  when  we  hear  a  certain  vowel  sound  the  ear  appreciates 
the  fact  that  it  has  a  certain  wave  form  as  well  as  amplitude 
and  wave  length,  for  we  distinguish  quality  in  sounds  as  well 
as  Iwulness  and  pitch. 

All  articulate  sounds  are  made  up  of  consonantal  sounds  and 
vowel  sounds.  The  latter  are  continuous  or  can  be  made  so 
to  be,  the  former  are  modulations  at  the  beginning  or  end  of  the 
vowel  sounds.  Thus  the  simplest  articulate  sound  is  a  syllable 
which  is  composed  of  a  vowel  sound  preceded  or  followed  by  a 
consonantal  sound.  Thus  the  word'P.4P^4  is  composed  of  two 
identical  syllables  PA,  each  composed  of  an  explosive  consonantal 
sound  indicated  by  the  P  and  followed  by  a  vowel  sound  Ah 
indicated  by  the  A. 

The  vowel  sound  is  made  up  of  the  sum  of  certain  simple  sine 
curve  aerial  vibrations  differing  in  phase  and  amplitude  with 
wave  lengths  or  frequencies  in  harmonic  relation. 

Accordingly,  if  we  are  to  transmit  intelligible  speech  by  tele- 
phone it  is  essential  that  the  broad  features  of  each  syllabic 
sound  shall  be  repeated  at  the  receiving  end.  This  means  that 
the  wave  form  of  the  current  which  emerges  from  the  line  at 
the  receiving  end  shall  not  be  extravagantly  different  from  the 


104        PEOPAGATION   OF   ELECTKIC   CUKRENTS 

wave  form  of  the  current  at  the  sending  end,  which  in  turn  must 
not  differ  greatly  from  the  wave  form  of  the  air  motion  in  front 
of  the  microphone  diaphragm. 

Hence  the  successful  transmission  of  speech  necessitates  thatthe 
various  constituent  harmonics  which  combine  to  make  the  wave 
form  of  the  current  at  the  sending  end  of  the  line  shall  he 
transmitted  so  that  they  are  not  much  displaced  in  relative 
phase  or  altered  in  relative  amplitude. 

4.  The  Reasons  for  the  Limitations  of  Tele- 
phony.— We  have  already  proved  that  the  speed  with  which  a 
simple  periodic  wave  of  electric  current  is  transmitted  along  a 
line  depends  upon  the  wave  length,  and  also  we  have  shown 
that  the  rate  at  which  the  amplitude  is  degraded  depends  also 
upon  the  wave  length  or  frequency. 

The  electrical  disturbances  of  short  wave  lengths  are  more 
rapidly  degraded  and  travel  faster  than  those  of  longer  wave 
length.  Hence  the  different  harmonic  constituents  into  which 
we  may  analyse  by  Fourier's  theorem  the  complex  wave  form  of 
the  line  current  representing  any  vowel  or  syllabic  sound  travel 
at  different  speeds  and  attenuate  at  different  rates  as  they  move 
along  the  line.  If  then  they  are  synthesised  by  the  ear  aided  by 
a  receiving  telephone  at  the  end  of  a  long  line,  the  result  may 
be  so  different  from  that  impressed  on  the  line  at  the  sending 
end  that  the  ear  may  no  longer  recognise  the  meaning  of  the 
sound.  This  change  in  the  wave  form  of  the  current  wave  sent 
along  the  line  as  it  travels  from  the  sending  to  the  receiving  end 
is  called  the  distorsion  due  to  the  line.  If  the  distorsion  is  not 
very  great  the  ear  recognises  the  articulate  sound  to  which  that 
current  wave  corresponds,  but  if  the  distorsion  has  proceeded 
beyond  a  certain  point  it  is  no  longer  recognisable.  The  process 
resembles  that  of  caricaturing  a  face.  The  caricature  is  a  draw- 
ing in  which  the  various  features  or  details  are  not  accurately 
drawn  but  distorted,  some  being  increased  or  decreased  more 
than  others.  If  the  process  has  not  been  carried  beyond  a 
certain  limit  we  still  guess  for  whom  it  is  meant,  but  beyond 
that  point  it  is  unrecognisable.  Hence  the  practical  limits  of 
telephony  are  found  in  this  distorsion  due  to  the  line.  Thus,  for 


TELEPHONY  AND  TELEPHONIC  CABLES   105 

instance,  with  a  certain  type  of  cable  we  may  obtain  excellent 
speech  transmission  over  twenty  miles,  good  over  thirty  miles, 
fair  or  not  very  bad  over  forty  miles,  but  extremely  bad  or 
impossible  over  sixty  miles.  In  this  matter  we  leave  out  of 
account  for  the  moment  all  questions  of  imperfection  of  the 
transmitter,  receiver,  speaker's  voice,  or  listener's  ear.  We 
assume  that  these  are  the  best  possible,  yet  nevertheless  the 
line  itself  by  reason  of  its  distorsion,  viz.,  by  the  unequal 
attenuation  and  velocity  of  simple  periodic  disturbances  of 
different  frequencies,  imposes  a  limit  on  the  distance  over  which 
good  speech  can  be  transmitted. 

The  improvement  of  telephony  is  therefore  bound  up  with  the 
improvement  in  the  qualities  of  the  line.  We  have  to  construct 
a  line  which  shall  be  non-distorsional  or  distorsionless,  or  at 
least  less  distorsional  than  existing  cables,  and  that  we  proceed 
to  discuss. 

5.  The  Improvement  of  Practical  Telephony. 

The  earliest  attempts  to  conduct  telephony  over  long  distances 
or  through  submarine  cables  brought  prominently  before  tele- 
phonists the  influence  of  the  line.  It  soon  became  clear  that 
both  resistance  and  capacity  in  the  line  were  obstacles  per  se 
to  long  distance  telephony  and  that  to  improve  it  the  resistance 
of  the  line  should  be  kept  low  and  its  capacity  small.  Hence 
aerial  lines  were  found  better  adapted  for  it  than  underground  or 
submarine  cables,  and  copper  wire  better  than  iron  wire.  It 
was  assumed  by  some  persons  imperfectly  acquainted  with 
electrical  theory  that  the  inductance  of  the  line  was  also  an 
obstacle  to  telephony.  A  little  knowledge  is  proverbially  a 
dangerous  thing.  Electricians  of  the  old  school,  educated  chiefly 
in  connection  with  continuous  currents  or  with  the  kind  of 
currents  required  in  slow  speed  telegraphy,  had  acquired  just 
sufficient  information  on  the  subject  to  know  that  the  inductance 
of  a  circuit  in  general  hinders  sudden  changes  in  the  current 
when  the  electromotive  force  is  suddenly  changed.  Hence  it 
was  but  natural  to  suppose  that  the  rapid  variations  of  current 
involved  in  telephony  would  also  be  resisted  by  the  inductance 
of  the  line.  Inductance  in  the  line  was  therefore  assumed  to  be 


106       PKOPAGATION   OF  ELECTRIC   CURRENTS 

detrimental  and  to  be  regarded  as  an  enemy  to  be  overcome. 
Moreover,  the  practicians. of  this  school  had  been  obliged  to 
master  some  elementary  knowledge  of  the  theory  of  the  sub- 
marine telegraph  cable,  which  will  occupy  us  in  a  later  chapter, 
and,  applying  this  without  hesitation  to  the  more  difficult  and 
different  problem  of  telephony,  had  come  to  the  conclusion  that 
the  great  remedy  for  the  difficulties  introduced  by  distributed 
capacity  in  the  cable  was  to  be  found  in  decreasing  the  resistance. 
Hence  an  empirical  rule  was  enunciated  which  endeavoured  to 
associate  good  telephony  with  less  than  a  certain  value  for  the 
product  of  the  capacity  and  resistance  per  mile  of  the  telephonic 
cable.  This  rule  was  commonly  called  the  "K  E"  law.  But 
accumulated  experience  showed  that  it  had  no  true  scientific 
basis  (see  Oliver  Heaviside's  work  "  Electromagnetic  Theory," 
Vol.  I.,  p.  321,  footnote).  The  problem  of  telephonic  transmission 
is  essentially  different  from  that  of  telegraphic  transmission. 

The  first  physicist  who  endeavoured  to  place  before  practical 
telephonists  a  valid  theory  of  telephonic  transmission  was  Mr. 
Oliver  Heaviside,  who  gave  the  fundamentals  of  the  right  theory 
in  a  paper  on  Electromagnetic  Induction  and  its  Propagation 
in  the  Electrician  in  1887,  Vol.  XIX.,  p.  79  (see  also  his  Collected 
Papers,  Vol.  II.,  p.  119).  He  also  published  in  The  Electrician  in 
1893  writings  of  considerable  originality  and  power  (see  issues 
for  July,  August,  September,  1893)  on  the  same  subject,  and 
these  were  collected  into  a  book  on  Electromagnetic  Theory 
(Vol.  L,  pp.  409—453),  published  in  1893. 

Meanwhile  the  conception  that  the  effects  of.  distributed 
capacity  could  be  annulled  by  inductance  or  leakage  had  arisen 
in  other  minds. 

Professor  S.  P.  Thompson  took  out  a  British  patent  (No.  '22,304) 
in  1891,  in  which  this  was  clearly  stated,  and  he  followed  it  by 
other  patents  in  1893  (Nos.  13,064  and  15,217),  in  the  specifica- 
tions of  which  he  describes  various  modes  of  carrying  the  idea 
out  in  practice.  Professor  S.  P.  Thompson  also  read  an  interest- 
ing paper  on  Ocean  Telephony  before  the  Electrical  Congress  at 
the  Chicago  World's  Fair  in  1893  which  attracted  considerable 
attention  to  the  subject,  in  which  the  methods  proposed  in  the 
above-mentioned  specifications  were  described,  and  the  general 


TELEPHONY  AND  TELEPHONIC  CABLES   107 

question  of  improving  telephony  and  telegraphy  discussed. 
Professor  Thompson  took  out  a  fourth  patent  (No.  13,581)  in 
1894. 

Mr.  Heaviside's  mathematical  investigations  had  led  him  to 
see  that  the  true  obstacle  to  long-distance  telephony  was  not 
capacity  or  inductance  in  themselves,  but  the  unequal  attenuation 
and  velocity  of  the  component  simple  periodic  waves  of  currents 
travelling  along  the  cable.  We  have  shown  in  Chapter  III.  that 
the  attenuation  of  a  simple  periodic  wave  of  current  travelling 
along  a  cable  is  dependent  upon  a  certain  quantity  a,  called  the 
attenuation  constant,  which  is  a  function  of  the  primary  constants 
of  the  cable  R,  C,  L,  and  S  and  of  the  frequency. 

The  amplitude  is  decreased  in  the  ratio  1  :  e~a  per  mile  of 
transmission.  Also  the  speed  W  with  which  the  wave  is  trans- 
mitted is  given  by  W  =  nk  =  p/(B,  where  n  is  the  frequency 
p  —  %-nn  and  /3  is  a  function  of  K,  C,  L,  S  and  p  called  the 
wave  length  constant.  Hence  waves  of  different  frequency  or 
wave  length  travel  at  different  speeds  and  attenuate  at  different 
rates. 

Now  Mr.  Heaviside  showed,  as  proved  in  Chapter  III.,  that 
if  the  primary  constants  of  the  cable  were  so  related  that 
CR=LS,  or  the  product  of  the  capacity  and  resistance  per  mile 
was  numerically  equal  to  the  product  of  the  inductance  and  leakage 
per  mile  in  homologous  units,  then  this  inequality  of  attenuation 
and  velocity  was  destroyed,  and  simple  periodic  waves  of  all 
frequencies  would  travel  on  such  a  cable  with  the  same  speed 
and  attenuation.  Also  the  wave  form  of  a  complex  wave  would 
travel  without  distorsion.  Hence  he  called  such  a  cable  a 
distorsionless  cable.  • 

The  reason  for  this  name  is  as  follows :  In  a  distorsionless 
cable  current  waves  of  all  frequencies  travel  along  the  cable  at 
the  same  speed,  viz.,  1/VCL,  and  attenuate  at  the  same  rate,  viz., 
are  reduced  in  amplitude  by  e~  ^8M  per  mile. 

Therefore  the  different  sine  curve  constituents  or  harmonics 
which  compose  a  current  wave  representing  any  given  vowel 
sound  are  not  relatively  altered  as  the  wave  proceeds.  In  other 
words,  the  wave  form  of  the  current  is  not  altered  in  form 
though  it  may  be  diminished  in  actual  size.  Hence  the  current 


108        PROPAGATION   OF  ELECTRIC   CURRENTS 

wave  arrives  afc  the  receiving  end  minified  or  reduced  in  scale, 
but  otherwise  a  fair  copy  of  that  which  set  out  from  the  sending 
end.  The  distorsion,  which  is  therefore  a  great  obstacle  to 
intelligibility,  is  cured  by  making  the  cable  have  such  constants 
that  CE  =  LS.  Since  in  all  ordinary  cables  the  value  of  CR 
is  much  greater  than  LS,  the  problem  of  making  a  cable  distor- 
sionless  is  capable  of  solution  in  many  ways.  For  example, 

(i.)  We  may  reduce  the  resistance  per  mile  R  to  the  necessary 
degree  of  smallness. 

(ii.)  We  may  decrease  the  capacity  per  mile  C. 

(iii.)  We  may  increase  the  inductance  per  mile  L. 

(iv.)  We  may  increase  the  leakage  of  the  cable  per  mile  S. 

(v.)  We  may  change  two  or  more  of  the  primary  constants  of 
the  cable  and  endeavour  to  make  the  product  CR  as  nearly  equal 
to  the  product  LS  as  possible. 

All  problems  in  engineering  are,  however,  ultimately  questions 
of  cost,  and  we  have  to  take  into  account  also  practicabilities  of 
construction  or  erection. 

It  was  long  ago  noticed,  however,  that  a  leak  in  a  telegraph 
or  telephone  line  was  not  always  a  detriment,  and  that 
distributed  leaks  sometimes  appeared  to  improve  telephonic 
speech. 

A  very  interesting  account  is  given  in  Mr.  Heaviside's  book 
"  Electromagnetic  Theory  "  (Vol.  I.,  pp.  420—433,  1st  ed.) 
of  the  effect  of  leaks  and  shunts  upon  telegraphic  and  telephonic 
transmission  in  certain  cases.  The  reader  would  do  well  to  refer 
to  this  account.  Mr.  Heaviside's  work  made  it  quite  clear  that 
inductance  up  to  a  certain  degree  in  a  telephone  line,  instead  of 
being  an  obstacle  to  long-distance  transmission,  was  the  tele- 
phonist's best  friend,  and  that  what  most  telephonic  cables 
required  to  improve  speech  through  them  was  not  less  but  more 
inductance.  He  discussed  in  a  general  manner  the  effect  of 
leaks  and  also  proved  that  these  were  in  certain  cases  an 
advantage. 

Mr.  Heaviside,  however,  did  not  reduce  his  general  principles 
to  such  detailed  instructions  as  to  compel  the  attention  of 
practical  telephonic  engineers.  Part  of  the  neglect  his  sugges- 
tions suffered  may  have  been  due  to  the  belief  that  though 


TELEPHONY  AND  TELEPHONIC  CABLES   109 

theoretically  correct  his  ideas  could  not  be  economically  carried 
into  practice,  and  that  a  more  practical  approach  to  improve- 
ment was  to  be  found  in  reducing  the  capacity  and  resistance  of 
the  line  rather  than  in  increasing  its  inductance.  About  the  same 
time  two  other  suggestions  were  made  by  Professor  S.  P.  Thomp- 
son, as  already  mentioned,  in  a  paper  on  Ocean  Telephony  read 
to  the  Electrical  Congress  meeting  in  1893  at  Chicago,  at  the 
World's  Fair  held  in  that  city.  In  this  paper  he  proposed, 
amongst  other  methods,  the  adoption  of  inductive  leaks  or 
shunts  across  the  cable  as  a  means  of  curing  the  distorsion. 
Again,  in  the  same  year,  Mr.  C.  J.  Reed,  following  one  of 
Professor  S.  P.  Thompson's  suggestions,  took  out  .United  States 
patents  (Nos.  510,612,  510,613,  December  12,  1893)  for  improve- 
ments in  telephone  lines  cut  up  into  sections  by  transformers. 
Professor  S.  P.  Thompson  urged  the  trial  of  his  method  in  his 
presidential  address  to  the  Institution  of  Electrical  Engineers  of 
London  in  1899.  Other  persons  also  either  suggested  or  patented 
methods  for  increasing  the  inductance  of  telephone  lines. 

Meanwhile  practical  telephonic  engineers  confined  their  efforts 
to  reducing  the  capacity  of  telephonic  cables,  and  as  far  as 
possible  consistently  with  economy  decreased  their  resistance 
by  the  use  of  heavy  high  conductivity  copper  wires  or  cables. 

A  considerable  reduction  in  capacity  in  underground  cables 
was  brought  about  by  the  introduction  of  paper  insulated  cables 
and  cables  called  dry  core  or  air  insulated  cables,  in  which  the 
copper  wire  was  loosely  wrapped  with  spirals  of  dry  paper 
sufficient  to  keep  the  wires  insulated  but  the  dielectric 
consisting  in  fact  of  air.  These  cables  were  then  lead  covered  to 
keep  them  dry.  In  long-distance  lines  and  cables  the  heaviest 
copper  conductor  was  adopted  consistent  with  economy. 

In  1899  and  1900  two  very  important  papers  were  published 
by  Professor  M.  I.  Pupin,  in  which  he  described  a  masterly 
investigation,  both  experimental  and  mathematical,  into  the 
properties  of  loaded  cables,  that  is,  cables  having  inductance  coils 
inserted  at  intervals  in  them. 

Pupin's  valuable  contribution  to  this  subject  was  the  proof 
given  by  him  that  a  non-uniform  cable  having  inductance  coils 
inserted  at  intervals  could  perform  the  same  function  as  a  cable 


110       PEOPAGATION   OF   ELECTRIC   CURRENTS 

of  equal  total  inductance  and  resistance,  but  with  the  inductance 
and  resistance  smoothly  distributed,  provided  that  the  wave 
length  of  the  electrical  disturbance  travelling  along  the  cable 
extended  over  at  least  nine  or  ten  coils. 

Pupin  was  thus  led  to  enunciate  a  suggestion  at  once 
scientifically  sound  and  practically  possible,  viz.,  to  improve 
telephonic  transmission  by  loading  the  cable  or  line  at  equidistant 
intervals,  small  compared  with  a  wave  length,  with  coils  of 
small  resistance  and  sufficiently  high  inductance. 

The  ideas  of  Heaviside  were  thus  extended  into  the  region  of 
practical  engineering,  and  Pupin's  loaded  cable  has  been  proved 
to  result  in  a  most  important  improvement  in  long-distance 
telephony. 

It  is  by  no  means  an  obvious  truth  that  a  number  of  separate 
inductance  coils  could  act  in  this  manner  to  improve  telephony. 
It  has  already  been  pointed  out  that  when  a  wave  of  electric 
current  or  potential  is  travelling  along  a  conductor,  if  it  arrives 
at  a  place  at  which  the  inductance  or  capacity  per  unit  of  length 
suddenly  changes,  there  will  be  a  reflection  of  part  of  the  wave 
just  as  in  the  case  of  a  ray  of  light  when  passing  from  one 
medium  to  another  of  a  different  refractive  index.  Accordingly 
an  inductance  coil  inserted  in  a  uniform  line  causes  a  loss  of 
wave  amplitude  by  reflection,  part  of  the  wave  being  transmitted 
through  the  coil  with  diminished  amplitude.  If  then  a  series 
of  such  coils  are  inserted  at  intervals  in  a  uniform  cable,  a 
series  of  reflections  may  take  place,  the  result  of  which  may  be 
to  immensely  diminish  the  amplitude  of  the  transmitted  wave. 

This  is  always  the  case  when  the  intervals  between  the  coils 
are  large  compared  with  the  wave  length  of  the  disturbance. 

If,  however,  the  wave  length  is  large  compared  with  the  length 
of  the  coil  intervals,  then  the  so  loaded  cable  acts  as  if  the  added 
inductance  were  uniformly  distributed. 

As  this  is  a  very  important  matter  we  shall  give  here  an 
analytical  proof  following  that  originally  given  by  Professor 
Pupin. 

6.  Pupin's    Theory    of   the    Unloaded    Cable. 

Pupin  prefaces  his  mathematical  treatment  of  the  problem  of 


TELEPHONY  AND  TELEPHONIC  CABLES    111 

the  loaded  cable  by  a  discussion  of  the  case  of  the  pro- 
pagation of  periodic  electric  currents  along  a  cable  of 
ordinary  type,  which  is  essential  for  the  sake  of  com- 
parison. In  the  following  discussion  we  shall  follow  Pupin's 
method  with  some  little  amplification  for  the  sake  of 
clearness.1 

Let  us  consider  a  cable  in  the  form  of  a  loop  (see  Fig.  4) 
having  an  alternator  A  at  the  sending  end  and  a  receiving 
instrument  B  at  the  receiving  end.  Let  the  alternator  generate 
a  simple  periodic  electromotive  force  which  may  be  represented 
as  the  real  part  or  horizontal  step  of  a  function  of  the  time 
denoted  by  E  €  '>'. 

Let  the  cable  have  per  unit  length  on  each  side  an  inductance 
L,  resistance  It,  and  capacity  with  respect  to  the  earth  C. 


FIG.  4. 

Let  distance  be  measured  from  the  alternator  and  let  the 
distance  between  the  alternator  and  receiving  instrument  be 
denoted  by  I.  At  distance  x  take  any  small  length  Bx.  Let  i 
be  the  current  in  the  cable  at  this  point.  Then  the  capacity  of 
this  length  with  respect  to  the  earth  is  CSx,  and  the  capacity 
with  respect  to  a  similar  element  in  the  return  half  of  the 

cable  is  „  C8x. 

If  then  v  is  the  potential  and  i  the  current  at  a  distance  x,  the 
potential  and  current  at  x  +  bx  are  v  -  ^~  bx  and  i  —  -r-  bx 
respectively.  Hence  the  fall  in  voltage  down  the  element  8x  is 


1  Pupin's  two  important  papers  are  to  be  found  in  the  Transactions  of  the 
AiiH'i'li'iui.  Institute  of  Electrical  KiHjlncers,  Vol.  XVI.,  p.  93,  1899,  and  Vol.  XVII., 
p.  4l.->,  19<)<).  The  first  is  entitled  "  Propagation  of  Line  Electrical  Waves"  (read 
March,  1899),  and  the  second  "Wave  Transmission  over  Non-uniform  Cables  and 
Long  Distance  Air-Lines  "  (read  May,  1900). 


112        PROPAGATION   OF   ELECTRIC   CURRENTS 

^  Bx  and  the  loss  in  current  is  -T-  Bx.      Hence  these  must   be 
equated  to  the  equivalent  expressions,  viz., 


~ 
at 

di  dv 


It  will  be  noticed  that  Pupin  considers  a  cable  without  leakage 
or  dielectric  conductance.  If  we  differentiate  the  first  of  these 
equations  with  regard  to  t  and  the  second  with  regard  to  x  to 
eliminate  v,  we  arrive  at  the  equation, 

_  d*i  .  T-,  di     1  d2i 


This  is  the  differential  equation  for  the  propagation  of  an 
electrical  disturbance  in  a  cable  having  inductance  L,  resistance 
R,  and  capacity  C  per  unit  length  of  both  lead  and  return 
separately,  the  leakage  being  negligible. 

To  formulate  the  boundary  conditions  we  assume  that  the 
alternator  has  a  resistance  7i0,  an  inductance  LO,  and  that  its 
capacity  is  equivalent  to  a  capacity  <70  in  series  with  its 
armature. 

Suppose  then  that  iQ  is  the  current  in  the  alternator  and  at 
the  sending  end  of  the  cable  and  that  v0  is  the  potential  difference 
of  the  two  sides  of  the  cable  at  the  sending  end. 

If  then  the  real  part  of  E  e^  represents  the  electromotive 
force  of  the  alternator,  the  potential  difference  r0  at  the  sending 
end  of  the  cable  is  the  difference  between  this  E.  M.  F.  and  the 
drop  in  voltage  down  the  alternator  circuit  and  the  capacity  in 
series  with  it. 

Hence  we  have  the  equation 

L^+Rfo+^i^t+v^EW.  .     (11) 

Again,  if  the  potential  difference  between  the  ends  of  the  cable 
at  the  receiving  end  is  v\  and  if  the  receiving  apparatus  is  equi- 
valent to  an  inductive  resistance  (Lb  EI)  in  series  with  a  capacity 
Ci  and  if  ii  is  the  current  at  the  receiving  end,  we  have  a  second 
boundary  equation,  viz., 

^-^  =  0     .        •        •    (12) 


TELEPHONY  AND  TELEPHONIC  CABLES   113 

If  the  E.M.F.  of  the  alternator  is  a  simple  periodic  function 
of  the  time,  then  after  a  short  time  the  current  at  all  parts  of  the 
line  will  also  be  proportional  to  e-K  Hence,  if  i  varies  as  e'pt, 

^-will  be  equal  tojpi  and  ~  equal  to  —  p*i. 

If  then  we  differentiate  equations  (11)  and  (12)  with  regard  to 
t  and  make  the  above  substitutions,  we  have 


.        .     (13) 

If  we  write          Mor  ^  (1-C0L0^H^CW    .         .        .     (14) 
and  DutoTJpCEcJpt        ....     (15) 


we  can  transform  (13)  into  the  equation 

dvn 

C^=D0-h0i0        ....     (16) 
ctt 

Now,  since  CBx  is  the  capacity  of  an  element  of  length  Bx  with 
regard  to  the  earth,  the  capacity  of  a  length  Bx  with  regard  to  a 

r\ 

similar  element  in  the  return  cable  must  be  -^  Bx,  and  hence  the 
fall  in  current  down  the  initial  element  Bx  at  the  sending  end 

which  is  expressed  by  —  ^  Bx  must  be  equal  to  -^  &%  -^j- 

or  ^~df~~  ^  rT         ....     (17) 

Making  the  substitution  in  (16)  we  have  as  the  boundary 
equation  at  the  sending  end 

-2  T^=D0-/i020       .        .        .        .     (18) 
Similarly  at  the  receiving  end 

*§=  -M,        ....    (19) 

We  have  next  to  consider  the  solution  of  the  differential  equa- 
tion (10).     A  solution  applicable  in  the  present  case  is 

where  AI  and  K%  are  functions  of  the    time  only  proportional 

tO   €./>'. 

•  It  is  easy  to  see  that  the  above  is  a  solution  provided  that 

—fjp  =  C(—p2L+jpR).        .        .        .    (21) 
B.C.  i 


114        PROPAGATION   OF   ELECTRIC   CURRENTS 

For  if  we  differentiate  (20)  with  regard  to  t  and  x  and  substitute 
in  the  original  equation  (10)  we  arrive  at  equation  (21). 

Since  —  ju2  is  a  complex  quantity  \L  is  also  a  complex  quantity, 
and  we  can  write  ^  =  3  +  ja=j  (a  —j(3). 


Hence  p+ja=  VCp  (pL-jR)        .        .        .     (22) 

or  p*-a*+j2ap=Cp(pL-jR). 

Therefore  p*-a*  =  LCp*} 

2«/3=  -CRp\ 

but  equating  the  sizes  of  the  vectors  in  (22)  we  have 

....     (24) 


and  from  (23)  and  (24)  we  arrive  at 

-     (25) 


Now,  since  (a  +  x)n  —  an  +  xnan~l  nearly,  when  x  is  small  com- 
pared with  a,  and  we  can  therefore  neglect  terms  involving  the 


_ 

square  and  higher  powers  of  x,  it  follows  that  ^/R'2  +p1Li*  =pL  -f 
when  pL  is  large  compared  with  R,  and  therefore  that 


Hence  when  pL/R  is  a  large  number  we  have 

_R     f(T) 

~^V_L  ....     (26) 

/3=p  V  CL  ) 

and  the  wave  velocity  W=  n\  =      _I 

V  CL 

Accordingly  the  attenuation  constant  a  and  the  wave  velocity  W 
are  independent  of  the  frequency  when  the  inductance  per  mile 
is  large  compared  with  the  resistance  per  mile  for  moderate 
frequencies. 

For  very  high  frequencies  pL  tends  to  be  always  greater  than 
R  under  any  circumstances. 

If  4  =  ^  Cos  /A  (/-»)+£•„  Sin  /A  (Z-ar)  .  .  .,  (27) 

it  follows  that  at  the  sending  end  where  x  =  0  and  i  =  io  we 

*Z        .        .     (28) 


TELEPHONY   AND    TELEPHONIC    CABLES        115 
Also  at  the  receiving  end  where  x  =  I  and  i  =  ii  we  have 

2  -T-1  =  -2Zow,  (29) 

ax 

but  by  (18)  -2-^  =  A-V'o   I 

...     (30) 
and  by  (19)  2  ^.=  _  Wl         j 

Also  from  (27) 
and  ^Cos^  +  ^Sm^j         _         ^         _     (31) 

Hence  from  (27),  (28),  (29),  (30)  and  (81)  it  can  easily  be  found 
that 


where 

^=(yi1-4/x2)Sin/zZ  +  2/*(/i0+7i1)Cos/^   .         .     (32) 

Accordingly  we  can  write  (27)  in  the  form 

(l-x)+h1$mp(l-x)}     .         .     (33) 

and  this  is  the  complete  solution  of  the  differential  equation  (10). 
When  7/o  =  hi  —  0  we  have 


0 
"2,*       Sin,** 

In  the  ahove  equations  /x  stands  for  fi+ja  where  a  is  the 
attenuation  constant  and  ft  the  wave  length  constant.  Hence 

the  wave  length  is  —  and  the  attenuation  for  a  distance  x  is  e""*. 

Equation  (33)  is  the  general  solution  of  the  differential 
equation  for  oscillations  either  free  or  forced.  If,  however,  the 
oscillations  are  free  oscillations,  then  D0  —  0  and  hence  in  this 
last  case  /x  must  have  such  a  value  as  to  make  F  =  0,  otherwise  i 
would  be  always  zero.  Accordingly  the  condition  for  free 
oscillations  is 

(7z0  h,  -  4  ^2)  Sin  pL  +  2  /x  (h0  +  hj  Cos  /x/  =  0  .        .     (35) 

Suppose  then  that  the  transmitting  and  receiving  apparatus 
are  removed  and  replaced  by  a  short  circuit.  This  is  equivalent 
to  assuming  Co  and  C\  both  to  be  infinitely  large.  Then  we 
have  7*o  =  hi  =  0. 

i  2 


116        PROPAGATION    OF   ELECTRIC   CURRENTS 

The  equation  (35)  then  reduces  to  Sin  \d  —  0,  and  hence  we 
must  have  ^  =  ~  where  s  is  some  integer  from  1  upwards. 

i 

S27T2 

Accordingly  —  /x2  =     —  p—  • 

Referring  to  equation  (21)  we  have 

-~    .  .     (36) 


If  we  write  k  for  jp  in  the  above  equation  it  becomes 

—         .         •        .        -     (37) 


Solving  this  quadratic  equation  we  have 


E  I    1       S27T2          W 


If  2L  is  large  compared  with  It,  then 


Hence  the  frequencies  of  the  possible  oscillations  are  obtained 
from  the  equation 


1    STT    I  1 
"=2.  TV  E 


EC  ' 

by  giving  s  various  integer  values.     The  velocity  of  propagation 

of  the  waves  is    W  =      .—  -,   and   hence    the     possible    wave 


lengths  are  the  values  of  2Z/s  for  various  integer  values  of  s, 
viz.,  2//1,  2//2,  2Z/8,  etc. 

In  the  next  place,  suppose  that  the  transmitter  has  no 
resistance  or  inductance  but  very  large  capacity,  and  that  the 
receiving  end  is  open.  Then  we  must  have  //o  =  0,  and 
/<i  =  infinity.  Equation  (35)  then  reduces  to  Cos  ^1  =  0  or 


/      2' 
where  s  is  any  integer.  , 

We  find  then  in  the  same  manner  as  in  the  former  case  that 
E 


//m 

-a? 

and  if  L  is  large  compared  with  R 

•  (42) 


TELEPHONY  AND  TELEPHONIC  CABLES   117 

and  the  wave  lengths  for  possible  free  vibrations  are  4//1,  4//3, 
41/5,  etc. 

7.  Pu pin's  Theory  of  the  Loaded  Cable.— In    the 

papers  previously  mentioned  Pupin  discusses  also  the  mathe- 
matical theory  of  the  cable  loaded  with  inductance  coils  at 
equal  intervals.  He  supposes  a  cable  to  have  coils  of  inductance 
L  and  resistance  E  inserted  at  equal  intervals  and  a  condenser 
of  capacity  C  to  be  connected  between  the  earth  and  the  junction 
between  each  coil.  Also  that  a  transmitter  having  inductance 
and  resistance  L0  and  It0  with  capacity  C0  is  placed  at  A  and  a 
receiver  with  similar  constants  Lb  Kif  C\  placed  at  B.  A  simple 


I— 

F  = 

—  »               — 

•••                                     •••• 

—  uvuu  — 

•••1                         ••• 
•••»                         •••• 

uuuu 

••••»  ••• 

•••1  •••• 

—  uuvv  ' 

•• 
••• 

L  = 

r™ 

IBB*                                           — 
•••»                                           •••• 

••••                         —  «— 
•••*                       aBBi 

—  'znnnp  — 

—  Pupiu  Arfcific 

••••  ^BBl 

••«•  •••• 

—  '07KNP  — 

ial  Cable. 

•••• 
•••• 

—  "fffffffl^- 

FIG.  5. 

periodic  electromotive  force  proportional  to  E  ^  is  in  opera- 
tion at  the  transmitter  end.  (See  Fig.  5.) 

The  conductor  thus  consists  of  Zn  coils  in  a  loop  with  2  (n — 1) 
condensers  to  earth  between.  The  whole  loop  is  thus  divided 
into  2  (n — 2)  component  circuits.  It  is  clear  that  when  n 
becomes  very  large  the  line  becomes  an  ordinary  cable.  The 
question  then  arises,  under  what  conditions  will  a  conductor  of 
this  kind  be  equivalent  to  a  uniform  cable  even  if  n  is  not 
infinitely  large  ?  The  problem  of  finding  the  time  of  electrical 
vibration  of  such  a  line  is  analogous  to  the  problem  of  finding 
the  free  vibrations  of  a  string  loaded  with  weights  at  equal 
intervals  which  was  solved  by  Lagrange  in  his  "  Mechanique 
Analytique  "  (Partie  VI.). 

Let  ii,  i2,  i3,  etc.,  be  the  currents  in  the  component  circuits  of 
the  loaded  line,  and  let  ri,  1'%,  r3,  etc.,  be  the  drops  in  potential 


118        PROPAGATION   OF   ELECTRIC   CURRENTS 

down  the  condensers.  Then  the  currents  through  the  condensers 
are 

gi=cf^,  </.=c§,'  etc., 

and  also  gi  =  ii  -  i^  r/2  =  i%  —  is,  etc.  Consider  then  in  the 
first  place  the  case  oi  forced  oscillations  in  such  a  loaded  cable. 
For  each  mesh  or  circuit  we  can  write  an  equation  as  follows : 


1st  circuit  ( 

2nd  circuit  L  ^+jR«a-ft?a—  t?i=0 

di 
(n  -  1)  th  circuit  L  —js-1  -f  -K^-i  +  vw  -i  -  v,t_2  =  0 

nth  circuit 


(43) 


When  the  steady  state  is  reached  the   currents  will   be   all 
simple  periodic  currents  and  proportional  to  eH 

Hence  for  -=-  .we  can  write  jp  and  for  -^  we  can  put  -  p2. 


The  above  equations  can  then  be  written 


.     (44) 


where 


,    .     (44a) 


Following  the  analogy  with  the  solution  of  the  differential 
equation  (10)  in  the  previous  section,  it  is  clear  that  a  solution 
of  the  equations  (43)  can  be  found  in  the  form 

im=K1  Cos  2  (n-m)  0+K2  Sin  2  (n-m)  6  .         .     (45) 

If  h  +  2  =  2  Cos  2  0,  then  all  the  equations  (44)  except 
the  first  and  last  will  be  satisfied  for  all  values  of  KI  and  K%. 


TELEPHONY   AND   TELEPHONIC   CABLES        119 

These  two  equations,  which  correspond  to  the  boundary  condi- 
tions in  the  case  of  the  uniform  cable,   will  be  satisfied  if 


*  Sin  20 

K  = 


^  Sin  2  (n  -  1)  6  -  4  Sin^  e  Sin  2w0  +  2  (fe0  +  h\)  Sin  <9  Cos  (2n  -1)0 
We  have  then  a  solution  for  im  in  the  form 

.    _  [2  Sin  6  Cos  (2/t-2//t  +  l)  0  +//i  Sin  2  Q-w)  0]  7?0 

lM*~Mi  Sin  2  (•»-!)  0-4  Sin*  0  Sin  2/<0+2  (7/0+/<i)  Sin  6  Cos  (2/4-1)  0    * 

0  is  a  complex  angle,  and  hence  forced  oscillations  of  a 
simple  periodic  type  on  a  non-uniform  cable  of  this  kind  are 
finally  simple  harmonic  damped  oscillations. 

Suppose  the  transmitter  and  receiver  absent,  and  the  cable 
short-circuited,  then  we  have  //0  =  hi  =  0,  and 


2  Sin  6  Sin  Zn  0 

In  the  next  place  let  us  consider  the  free  oscillations. 

The  expression  for  the  current  given  in  equation  (47)  must 
hold  for  free  as  well  as  forced  oscillations.  When  the  oscillations 
are  free,  then  the  E.M.F.  of  the  transmitter  is  zero,  and  hence 
DQ  =  0.  Accordingly  the  denominator  of  (47)  must  then  be 
zero  to  prevent  the  current  vanishing. 

Hence  we  must  have  in  the  case  of  free  oscillations 
//.„  hi  Sin  (2;i--2)  (9-4  Sin  2<9  Sin  2?i  <9  + 

2  (Vhfei)Sin0Cos  (2n-l)  (9-0          .         .     (49) 

The  first  important  case  to  consider  is  when  the  transmitter 
and  receiver  are  absent,  and  the  cable  short-circuited  at  both 
ends.  Then  /*<,  =  hi  =  0  and  im  =  B  Cos  (2/i  --  2wi  +  1)  6. 
If  in  equations  (44)  we  substitute  the  values  of  cji  =  ii  —  i2, 
02  =  f2  —  ?3,  etc.,  we  have 


*„-*„.;-  0=  -M 

Now  it  is  found  from  (49)  that  the  value 


is  a  solution  of  the  differential  equations  (50)  for  7/0  =  //i  =  DO  =  0 


120        PROPAGATION   OF  ELECTRIC   CURRENTS 

provided  that  0  —  5-,    where  s  is  some  positive  integer  from 

1  to  2w. 

Hence  the  most  general  solution  for  the  current  is  then 

STT 


'   (51) 

Also  im  is  a  periodic  function  of  the  time,  and  may  be  written 

V    -  -    "S      7T  cr>at  /£|9\ 

^m—    *     A8  e   a  .  .  .  .      (M) 

s=l 

Hence  in  (51)  each  amplitude  contains  the  factor  e^' 

The   constant  ps,    which    determines    the    period    and    the 

damping,  is  determined  as  follows  : 
From  the  second  equation  in  (50)  we  have 


Now  im  varies  as  Cos  (2ra  —  2m  +  1)  6.  Hence,  giving  m 
values  1,  2,  3,  successively,  we  have 

i:  :  iz :  i3  =  Cos  (2w  — 1)  0  ;  Cos  (2w  — 3)  0  :  Cos  (271-5)  <9 

7   ,  0     Cos  (271-1)  0+Cos  (271-5)  6 
and  ^+2  =  Cos  (2,1-3)0 

The  quantity  on  the  right-hand  side  is  equal  to  2  Cos  2  6. 
Hence  h  =  2  Cos  2(9  -  2  =  -  4  Sin  2<9. 
Hence  for  free  oscillations  we  have 

h=p*  LC+p8  BC=  -4  Sin2  0=  -4  Sin2  ^         .     (53) 

Before  solving  the  equation  (53)  it  is  desirable  to  make  the 
following  substitutions : 

Let  I/',  Rr,  and  Cr  be  the  total  inductance,  resistance,  and 
capacity  of  one  half  of  the  loaded  conductor.  Then 

L=^E=K,C=91. 

n'          n*          n 

Let  I  denote  the  distance  between  the  ends  or  half-length  of  a 
line  having  inductance,  resistance,  and  capacity  per  unit  of 
length  denoted  by  u,  r,  and  c,  and  let  this  uniform  line  have 
such  values  that 

lu=L',  lr=B',  k  =  C'. 


TELEPHONY  AND  TELEPHONIC  CABLES    121 

This   uniform    line   will   be   called   the    corresponding    uniform 
conductor. 

We  can  then  write  the  equation  (53)  in  the  form 

^(P'2uc+P*cr)=  -4Sin'2^    •         •         •     (54) 
where  ps  takes  the  place  of  jp  in  equations  (44a). 
Solving  this  quadratic,  we  have 


or  P,=  ~±,. 

If  u  is  large  compared  with  r  we  have 

.   2tt   Q.          S7T        /T 

A-'V      2SV^' 

and  the  possible  frequencies  /8  are  given  by 


. 

The  equation  for  the  current  can  then  be  written 

r      s=2n  P_ 

*'Bl  =  e-§5'    2    .4,  Cos  (2ra-2w+l)  ~-  Cos  (kj-fy       .     (57) 

S=:l  ^^ 

The  oscillations  in  the  non-uniform  cable  have  therefore  the  same 
damping  coefficient  as  those  in  the  equivalent  uniform  conductor. 

The  second  important  case  is  when  the  transmitter  end  of  the 
cable  is  short-circuited  and  the  receiver  end  is  open.  Then  we 
have  //o  =  0,  hi  =  oo  and  D0  =  0. 

Accordingly  from  equation  (47)  we  find  that  then 

im=B  Sin  (2n—^m)  6, 

provided  also  that  Cos  2  (n  —  1)  6  —  0  to  make  the  denominator 
of  (47)  always  zero. 

Hence  6  can  have  the  values 


and  therefore,  as  in  the  other  case,  the  possible  frequencies  /, 
are  given  by  the  equation 

In     .    2s+l  TT       T 


and  the  current  by 

r      s=2n 

im  =  *-*S    S    A.2u^  Sin  (2w-2w  +  2)  Cos 


122        PEOPAGATION   OF   ELECTRIC   CURRENTS 


The  angles  ^and      —i  \  nave  a  definite  physical  meaning. 

If  we  consider  the  sth  harmonic  oscillation,  then  the  current  at 
the  mth  coil,  which  is  denoted  by  (im)M  is  given  by 

(O.  =  A  Cos  (2w-2w+l)  |^Cos  (k.t-4). 
The  current  at  the  with  coil  is  also 

(V).=^.  Cos  (2^-2^+1)  ~  Cos  (ktt-4>)' 

If  these  coils  are  one  wave  length  apart,  then  (im)8  =  (*,ni)s,  and 
mi  —  ??i  is  the  number  of  coils  covered  by  one  wave.  But  then 
we  must  have 


Hence  mi  —  n  =  —  =  />      and    this  last  expression  is  there- 

o 

fore  the  number  of  coils  covered  by  one  wave  length  of  the  sth 
harmonic. 

In  the  second  case  it  can  be  shown  in  a  similar  manner  that 


A  4-         V     .•  £     S7r  1      2S  +  1   7T  1    2?T 

Accordingly  instead  of       and      ~~  we  can  wn^e         * 


If  we  consider  27r  to  represent  the  wave  length  and  y  the  angle 
which  is  the  same  fraction  of  2??  that  the  distance  d  between 
two  consecutive  coils  is  of  a  wave  length,  then  2-Tr  :  y  =  \  :  d,  and 
therefore  ZTT/VS  =  y. 

1  TT          Sir         -   _..     1  „.       STT 

Hence  3  7  ==  -  =  ^  and  Sin  g  y  =  Sin  ^. 

Now  on  comparing  equation  (40)  for  the  frequency  of  free 
oscillations  in  a  uniform  cable  with  equation  (56),  which  gives 
the  same  quantity  for  the  non-uniform  loaded  cable,  it  is  clear 

that  if  the  coils  are  so  close  that  o  7  is  practically  the   same 

as  Sin  -&  y,  then  the  loaded  line  has  free  vibrations  like  the 

equivalent  equally  loaded  cable.  Accordingly  Pupin  reduced 
the  solution  of  the  problem  to  a  verbal  statement,  which  may  be 
called  Pupin's  Law,  as  follows  : 


TELEPHONY   AND   TELEPHONIC   CABLES        123 

If  there  be  a  non-uniform  cable  line  loaded  with  inductance 
coils  at  equal  intervals,  and  if  we  consider  the  total  inductance 
and  resistance  to  be  smoothly  distributed  along  the  line,  then 
these  two  lines,  the  non-uniform  and  uniform  lines,  having  the 
same  total  resistance  and  inductance,  will  be  electrically  equiva- 
lent for  transmission  purposes  as  long  as  one  half  of  the  distance 
between  two  adjacent  coils  expressed  as  a  fraction  of  2w  taken 
as  the  wa've  length,  is  an  angle  so  small  that  its  sine  has  practi- 
cally the  same  numerical  value  as  that  angle  in  circular 
measure. 

Thus,  for  instance,  if  there  are  ten  coils  per  wave  the  angular 
distance  of  two  successive  coils  is  36°,  and 


But  Sine  18°  =  0*3090,  and  therefore  *  y  exceeds  Sin  »  y  by  1*6%. 
If  there  are  five  coils  per  wave,  then  o  7  —  36°  —  0*628  radian  ; 

Zi 

and  Sin  2  y  —  Sine  36°  =  0*588. 
Here  ^  y  exceeds  Sin  ^  7  by  6*8%. 
If  there  are  four  coils  per  wave,  then  ^  y  —  45°  ==  0*785 

radian,  whilst  Sin  ^  y  —  Sine  45°  =  0*707,  and  \  y  exceeds  Sin  ^  y 

by  nearly  11%. 

Accordingly  it  is  clear  that  if  there  are  at  least  nine  coils  per 
wave  the  non-uniform  cable  is  for  that  frequency  practically 
equivalent  to  a  cable  in  which  the  same  inductance  and  resistance 
is  smoothly  distributed. 

Pupin  then  shows  in  the  papers  mentioned  that  the  same  law 
holds  good  for  forced  as  for  free  oscillations  and  -also  for  a  cable 
in  which  capacity  is  added  in  series  with  each  loading  inductance 
coil. 

Pupin  was  therefore  led  to  a  very  practical  solution  of  the 
problem  of  constructing  a  telephone  line  which,  if  not  absolutely 
distorsionless,  was  at  least  much  less  distorsional  than  ordinary 
unloaded  lines. 


124        PROPAGATION   OF   ELECTRIC   CURRENTS 

Consider,  for  instance,  the  National  Telephone  Company's 
standard  line,  viz.,  a  telephone  cable  having  a  resistance  of 
88  ohms  per  loop  mile,  an  inductance  of  0*001  henry  per  loop 
mile,  a  capacity  of  '05  microfarad  per  loop  mile,  and  no  sensible 
leakage.  Then  E  =  88,  C  =  '05  X  10~6,  L  =  O'OOl,  S  =  0. 

Therefore  for  this  cable  /3  =   y  -j-  |  ,J'R*+p*L*+Lp\  where 

p  =  %TT  times  the  frequency. 

As  regards  the  frequency  or  range  of  frequency  employed  in 
telephony,  the  actual  frequencies  of  the  simple  periodic  oscilla- 
tions with  which  articulate  sounds  may  be  analysed  vary 
between  100  and  2,000  or  so.  It  has  been  found,  however,  that 
a  mean  value  of  about  800  may  be  employed  in  the  formuhe  for 
the  attenuation  and  wave  length  constants,  or  in  round  numbers 
we  may  take  p  =  5,000  for  the  case  of  articulate  speech.  Put- 
ting, then,  p  —  5,000  in  the  above  formula,  we  have  pL  =  5, 
p  C  -  25  X  10~5,  and 


Hence  we  have  (3=  V12-5x93-lxlO-5=0-108. 

Therefore  A  =  27T//3  =  58'2  miles. 

The  wave  length  for  the  frequency  of  about  800  is  therefore 
nearly  60  miles.     Also  the  attenuation  constant  a  is 
A/12-5  x  83-1  x!0-5  =  0-102. 

Suppose  then  that  the  above  cable  has  inserted  in  it  every  two 
miles  a  loading  coil  or  inductance  coil  having  an  inductance 
of  0*2  heavy  and  negligible  resistance.  Then  the  inductance 
per  mile  becomes  O'l  henry,  and  for  the  loaded  line  and  same 
frequency  we  have  E  =  88,  L  —  0*1,  C  =  5  X  10~8,  p  =  5000. 
Hence  p  L  =  500  p  C  =  25  X  10~5.  Therefore 


v/7744  +  25-104-500  [=0-031, 


OK         (•  \ 

2W\  V  7744 +  25-10H  500  j -=0-354, 

and  A  =  ~  =  18  nearly. 

Accordingly  the  effect  of  loading  is  to  reduce  the  original  attenua- 
tion  constant   to    q-  and    the    wave  length  in  the  same   ratio. 


TELEPHONY  AND  TELEPHONIC  CABLES    125 

Since  there  is  one  loading  coil  every  two  miles,  and  since  the  wave 
length  of  the  loaded  line  is  18  miles,  it  follows  that  there  are  nine 
coils  per  wave  length  of  the  loaded  line.  Hence  the  inter-coil 
distance  is  short  compared  with  the  wave  length.  It  is  found 
that  under  these  conditions  the  loss  by  reflection  at  each  coil  is 
not  serious.  If,  however,  the  inter-coil  distance  were  large 
compared  with  the  wave  length,  the  loss  of  wave  energy  at  each 
reflection  would  be  considerable.  We  have  already  shown  in 
Chapter  III.  that  when  a  wave  of  current  passes  across  a  point 
which  marks  a  change  in  the  constant  of  the  line,  say  a  sudden 
variation  of  inductance  per  mile,  then  reflection  occurs,  part  of 
the  wave  being  transmitted  and  part  reflected.  If  this  process 
is  repeated  at  intervals  long  compared  with  the  wave  length  the 
wave  energy  is  soon  frittered  away.  Hence  if  the  wave  form  is 
complex  and  if  it  passes  over  a  line  loaded  with  lumps  of 
inductance  placed  at  intervals  which  are  short  compared  with 
the  fundamental  wave  length,  but  long  compared  with  the  higher 
harmonic  wave  length,  then  the  effect  will  be  to  stop  these  latter 
or  filter  out  the  harmonics  and  let  pass  only  the  fundamental 
sine  curve  component. 

Hence  any  sudden  change  in  the  capacity  or  inductance  per 
mile  is  a  source  of  energy  loss  to  the  transmitted  wave  owing 
to  a  reflection  of  part  of  the  wave  at  this  surface.  An  analogous 
effect  is  produced  in  the  case  of  light.  Suppose  a  tube  down 
which  a  ray  of  light  is  sent.  Let  a  partition  of  glass  be  placed 
in  the  tube.  Then  at  this  point  there  is  a  sudden  change  in 
the  refractive  index  of  the  medium.  Accordingly  part  of  the 
wave  is  transmitted  and  part  reflected  back.  If  we  were  to 
place  many  plates  of  glass  in  the  tube  separated  by  intervals 
large  compared  with  a  wave  length  there  would  be  a  loss  of 
light  at  each  reflection,  and  the  wave  would  pass  through 
considerably  weakened  by  the  reflections. 

If  the  thickness  of  the  plates  and  of  the  interspaces  were  short 
compared  with  the  wave  length  this  would  not  occur. 

Pieturning  then  to  the  above-mentioned  standard  cable  when 
unloaded  and  loadeJ,  it  is  clear  that  for  the  unloaded  cable  the 
propagation  constant  P  =  a  -\-jfi  is  a  vector 

P  =  0-102 +/  0-108  -0-149  /45° 


126   PROPAGATION  OF  ELECTRIC  CURRENTS 

nearly,  whereas  after  loading  the  cable  the  propagation  constant 
becomes  P'  =  a'  +  jfir,  or  is  a  vector 

P'  =  0-031  +j  0-354  =  0-356  /85°. 

Hence  the  loading  not  only  increases  the  size  of  the  propaga- 
tion constant,  but  increases  its  slope. 

Accordingly  in  this  cable  after  loading  every  two  miles  the 
wave  length  is  18  miles  and  there  are  nine  coils  per  wave. 
The  wave  velocity  W  —  \\ VCL  before  loading  is  nearly  143,000 
miles  per  second,  but  after  loading  it  is  reduced  to  14,300 
miles  per  second,  or  about  7,000  coils  would  be  passed  through 
per  second. 

Again,  since  ZQ,  the  initial  sending  end  impedance,  is  equal 

to  ^  :-,  the  result  of  loading  the  cable  is  to  increase  Z0, 

v  K+jpC 

and  this  decreases  the  current  into  the  sending  end  for  a  given 
impressed  E.M.F.  Accordingly  we  see  that  loading  the  cable 
has  the  effect  of  producing  five  great  improvements,  as  follows : 

1.  It  increases  the  value  of  the  propagation  constant  P   both 
as  regards  size  and  slope. 

2.  It  reduces  the  value  of  the  attenuation  constant  a. 

3.  It  reduces  the  wave  length    A  for  a  given  frequency  and 
also  the  wave  velocity  W. 

4.  It  gives  the  cable  a  larger  initial  sending  end  impedance, 
and  therefore  reduces  the  current  into  the  cable  with  a  given 
impressed  voltage. 

5.  It  tends  to  unify  or  equalise  the  attenuation  constants  and 
also  the  wave  velocities  for  different  frequencies. 

The  result  is  that  the  wave  form  is  propagated  not  only  with 
less  attenuation,  but  with  less  distorsion  or  loss  of  individuality, 
owing  to  the  more  equal  attenuation  and  velocity  of  the  various 
harmonic  constituents. 

8.  Campbell's  Theory   of  the   Loaded   Cable. 

As  long  as  the  loading  coils  are  placed  at  such  intervals  that 
there  are  eight  or  nine  coils  per.  wave  length  calculated  on  the 
assumption  that  the  added  inductance  is  smoothly  or  uniformly 
distributed,  experience  shows  that  the  so  calculated  attenuation 
constant  agrees  with  the  results  of  experiment. 


TELEPHONY  AND  TELEPHONIC  CABLES    127 

It  is,  however,  necessary  to  establish  a  more  general  theory  of 
the  loaded  line  and  to  show  how  the  propagation  constant  P, 
attenuation  constant  a,  and  wave  length  constant  /3  can  be 
calculated  from  the  values  of  the  primary  constants  of  the  line 
when  unloaded  and  from  the  inductance  and  resistance  of  the 
loading  coils  and  their  distance  apart,  knowing  of  course  the 
frequency.  A  general  theory  of  the  loaded  line  has  been  given 
by  Mr.  G.  A.  Campbell.1 

In  the  paper  in  which  he  gives  the  theory  Campbell  assumes 
that  the  line  is  of  very  considerable  length  and  is  loaded  at 
intervals  of  distance  equal  to  d  with  coils  of  impedance  Z. 


FIG.  6. 

A  diagrammatic  representation  of  the  line  is  as  shown  in. 
Fig.  6. 

The  distance  d  is  measured  from  the  centre  of  one  loading 
coil  A  to  the  centre  of  the  next  coil  B,  and  the  impedance  Z  of 
each  coil  is  the  sum  of  the  two  parts  in  the  lead  and  return 
respectively. 

If  the  line  is  very  long  we  may  assume  that  the  average 
propagation  constant  is  the  same  as  the  average  propagation 
constant  of  one  single  section  of  length  d,  comprising  the  two 
half  loading  coils  at  each  end  and  the  length  of  line  between 
them.  The  length  of  this  section  of  line  will  always  be  very 
long  compared  with  the  length  of  a  loading  coil. 

Furthermore  we  may  assume  that  in  the  loading  coil  itself  the 
current  is  the  same  at  all  parts  of  the  wire  composing  it,  and 
therefore  the  same  at  the  centre  as  at  the  end. 

We  can  then  imagine  a  short  circuit  made  at  the  centre  of  one 

1   Sec  /V//V.  .)/,/,/..  Vol.  V.,  p.  319,  March,  11)03. 


128        PROPAGATION   OF   ELECTRIC   CURRENTS 

coil  B  so  that  the  current  at  the  centre  of  that  coil,  which  we 
shall  call  72,  remains  the  same  as  before.  Also  we  can  imagine 
such  an  electromotive  force  applied  between  the  centres  of  the 
two  parts  of  the  coil  A  that  the  current  there  retains  the  same 
value  Ii.  Hence  the  current  in  all  parts  of  the  section  AB  of 
the  infinite  line  remains  the  same,  and  we  can  suppose  that  the 
parts  of  the  line  beyond  B  and  before  A  are  removed.  We  have 
then  simply  to  find  the  average  propagation  constant  of  this 
finite  line  to  solve  our  problem.  Following  a  suggestion  of 
Dr.  A.  E.  Kennelly,  we  may  regard  this  finite  line  in  one  of  two 
ways  :  — 

(i.)  As  a  line  of  propagation  constant  P,  which  is  the  same  as 
that  of  the  unloaded  line  or  lengths  of  line  between  the  coils, 
which  is  closed  at  the  receiving  end  through  a  receiving 
instrument  of  impedance  Z/2. 

(ii.)  We  may  regard  the  line  as  one  having  an  average  propa- 
gation constant  Pr,  which  is  short-circuited  at  the  receiving  end. 

In  both  cases  the  line  itself  is  assumed  to  have  the  same 
initial  sending  end  impedance  ZQ. 

If  then  the  current  at  the  sending  end  is  Ii  and  that  at  the 
receiving  end  is  /2,  we  have  already  shown  (see  Chapter  III., 
equation  (60))  that  in  a  line  of  initial  sending  end  impedance  Z0 
and  having  a  receiving  instrument  of  impedance  Zr  at  the  end 
the  currents  /i  and  1%  are  related  as  follows  : 


=  Cosh  PI    +        SinhPZ  (60) 

^2  ^o 

In  the  present  case  the  length  of  line  is  d,  and  the  propagation 
constant  is  P,  and  the  impedance  of  the  supposed  receiving 
instrument  is  Z/2. 

Hence  we  have  then 

^  =  Cosh  Pd+<Hr  Sinh  Pd     .  (61) 

•*a  ^^  o  . 

Again,  we  have  shown  (see  Chapter  III.,  equation  (49))  that  in 
the  case  of  a  line  of  length  d  and  average  propagation  constant 
P',  which  is  short-circuited  at  the  receiving  end,  the  ratio  of  the 
currents  is  given  by 

^  =  Cosh  Pel  (62) 


TELEPHONY  AND  TELEPHONIC  CABLES    129 

Hence  this  applies  to  the  case  (ii.).     Equating  these  values  of 
Ii/I*)  we  have 

n> 

Cosh  P'd  =  Cosh  Pd  +  n^  Sinh  Pel  (63) 

^ZQ 

The  above  equation  is  that  given  by  Mr.  Campbell  (see  Phil. 
Mag.,  Vol.  V.,  p.  319,  1903),  but  the  process  of  reasoning  by 
which  he  arrives  at  it  is  based  upon  a  consideration  of  the 
coefficients  of  reflection  and  transmission  of  each  coil.  His 
argument  is  much  more  difficult  to  follow  than  that  given  above, 
and  in  the  opinion  of  the  author  contains  one  small  inconsistency 
between  his  lettered  diagram  and  the  text  which  is  extremely 
puzzling.  Accordingly  we  shall  not  reproduce  his  proof 
verbatim  here,  but  leave  the  reader  to  consult  the  original 
paper. 

We  can  put  Campbell's  equation  into  another  form. 

a> 

If  we  denote  ^-  by  tanh  y,  as  before,  we  have 

A<&0 

Cosh  P'd  =  Cosh  Pd+  tanh  y  Sinh  Pd         .         .     (64) 
which  can  be  written 


(66) 


We  have  already  given  the  expressions  for  calculating  the  value 
of  an  inverse  hyperbolic  function  such  as  Cosh"1^  or  Sinh"1^. 
Hence  if  P,  d,  and  y  are  given,  we  can  reduce  the  value  of 

Cosh  (PcZ+y)/Coshy 
to  the  form  x  +  jy,  and  we  have  then  for  the  value  of  Pr  =  a'  -\-j(3r 

F=lcosh-i(a;+jy)   ....    (67) 

(Ju 

But  this  last  is  a  vector  quantity,  and,  in  accordance  with  the 
proof  given  at  the  end  of  Chapter  L,  can  be  written  in  the  form 


Hence,  equating  horizontal  and  vertical  steps,  we  have  for  the 
B.C.  K 


130        PROPAGATION   OF   ELECTRIC   CURRENTS 

value  of  the  average  attenuation  constant  a'  of  the  loaded  line 
the  expression 

3!±1L'          .     (69) 


and  for  the  average  wave  length  constant 


^  ±  .         (70) 

Cl  A 

The  above  formulae  lend  themselves  without  difficulty  to 
numerical  calculation,  but  require  some  care  in  use.  They  enable 
us  to  calculate  the  attenuation  constant  for  a  line  of  certain 
known  primary  constants  loaded  at  intervals  of  distance  d  with 
inductance  coils  of  impedance  Z. 

On  the  other  hand,  when  the  coils  are  spaced  apart  so  closely 

that  the  distance  d  does  not  exceed  5  TT,  or  one-ninth  of  a  wave 

y  p 

length  on  the  loaded  cable,  then  we  can  obtain  just  as  good  a 
value  for  ar  and  ft'  by  considering  the  inductance  of  the  coils 
smoothly  distributed  along  the  line. 

If,  however,  the  coils  are  fewer  than  about  nine  per  wave  length, 
then  the  resultant  or  true  attenuation  constant  of  the  loaded 
line  is  greater  than  that  calculated  on  the  assumption  that  the 
added  inductance  is  smoothly  distributed  over  the  line. 

Let  a'  be  this  true  attenuation  constant  and  a"  the  attenuation 
constant  calculated  from  the  assumption  of  uniformly  distributed 
inductance,  and  let  ft'  and  ft"  and  A'  and  A"  be  the  corre- 
sponding wave  length  constants  and  wave  lengths. 

Suppose  that  an  unloaded  line  has  a  resistance  of  R  ohms  and 
an  inductance  of  L  henrys  per  mile,  the  inductance  being  very 
small.  Let  this  line  be  loaded  with  impedance  coils  such  that 
the  total  added  resistance  makes  the  line  equivalent  to  one  having 
R  +  R'  ohms  per  mile  and  the  total  inductance  equal  to  a  line 
of  L  +  L1  henrys  per  mile. 

Then  these  values  of  the  total  resistance  and  inductance  may 
be  used  as  the  R  and  L  in  the  formula  for  calculating  the 
attenuation  and  wave  length  constants,  and  they  give  us 
respectively  the  values  of  a"  and  ft". 

Suppose  then  that  R'  is  given  such  a  value  that  it  is  about 
equal  to  J£/2,  then  the  attenuation  constant  a",  calculated  from 


TELEPHONY  AND  TELEPHONIC  CABLES    131 


the  smoothly  distrihuted  resistance  and  inductance,  is  nearly 
equal  to  the  true  attenuation  constant  a'  when  there  are  nine  coils 
per  wave.  If,  however,  there  are  less  coils  per  wave,  then  a'  is 
greater  than  a"  by  a  certain  percentage,  as  shown  in  the  table  below. 


Number  of  coils  per 
wave  length  A.". 

Distance  between 
coils  =  d. 

Percentage  by  which 
a  exceeds  a". 

9 

X"/9 

Practically  zero. 

8 

A'  78 

1% 

7 

X"/7 

2% 

6 

X"/6 

3% 

5 

X"/6 

7% 

4 

X"/4 

16% 

3 

X"/3 

200% 

The  results  vary  somewhat  with  the  ratio  of  R '/R  and  L'/L 
In  any  case  for  less  than  four  or  five  coils  per  wave  the  actual 
attenuation  is  very  much  greater  than  the  attenuation  calculated 
on  the  assumption  that  the  added  inductance  and  resistance  are 
smoothly  distributed. 

If  we  have  as  few  as  three  coils  per  wave  the  attenuation 
becomes  so  large  that  we  may  say  that  practically  the  line  will 
not  pass  such  a  wave  length  at  all. 

Suppose  that  there  are  N  impedance  coils  in  the  length  of  line 
which  the  current  wave  travels  over  per  second ;  and  let  these 
coils  be  separated  by  a  distance  d. 

Then  Nd  is  the  distance  travelled  by  the  wave  per  second, 
which  is  the  same  as  its  velocity,  W. 

But  the  wave  velocity  W  =  n\,  where  n  is  the  frequency  and 
A  is  the  wave  length.  Hence  we  have 

Nd=W=n\, 

N=n^. 

If  we  take  n  =  800  as  an  average  value  of  the  frequency  in 
articulate  speech,  then,  since  experiment  shows  that  a  value  of  \jd 
equal  to  9  gives  good  results,  we  have  N  —  800  X  9  =  7,200. 
In  other  words,  the  rate  of  load  traversing  is  7,200  coils  per 
second. 

K  2 


132        PKOPAGATION   OF   ELECTEIC   CURRENTS 

Experiment  shows  also  that  \/d  cannot  practically  be  less  than 
4  or  3.  Hence  7,200/3  —  2,400  is  the  highest  frequency  we 
can  be  concerned  with  in  practical  telephony. 

For  such  a  rate  of  load  traversing  and  for  such  frequencies  we 
can  consider  that  the  unequally  distributed  impedance  at  the  rate 
of  nine  coils  per  wave  gives  us  a  line  which  is  for  all  practical 
purposes  an  equally  or  smoothly  loaded  line  of  approximately 
distorsional  character. 

Thus,  for  instance,  if  a  line  having  90  ohms  per  mile  resistance 
andO'OOl  henry  inductance  and  0!05  X  10~6  farads  capacity  had 
inductance  coils  of  approximately  0*2  henry  inductance  and 
20  ohms  resistance  inserted  every  two  miles,  this  would  be 
equivalent  to  adding  10  ohms  and  O'l  henry  per  mile  ;  then  the 
total  resistance  would  be  100  ohms  per  mile,  and  the  product 
CR  per  mile  would  be  equal  to  5  X  10~6.  Hence,  if  the  insula- 
tion resistance  were  reduced  to  20,000  ohms  per  mile,  we  should 
have  S  =  5  X  10~5  and  LS  =  5  X  lO"6. 

Such  a  line  would  be  theoretically  distorsionless  in  that  all 
wave  frequencies  would  travel  along  it  at  the  same  rate.  The 
attenuation  constant  a'  would  be  approximately  equal  to  0'07, 
whereas  that  of  the  unloaded  line  would  be  at  least  O'l. 

These  explanations  will  suffice  to  show  the  very  great  improve- 
ment that  is  made  in  the  transmission  properties  of  a  telephone 
line  by  suitable  loading  with  impedance  coils,  and  that,  provided 
the  insulation  is  not  too  good,  we  can  approximate  to  the 
properties  of  a  distorsionless  line. 

9.  Other  Methods  of  reducing  the  Distorsion 
of  Telephone  Lines. — In  addition  to  the  method  above 
explained  of  loading  the  line  with  impedances,  two  other 
methods  have  been  suggested  for  overcoming  the  distorsional 
quality  of  a  telephone  cable.  One  of  these,  due  to  Professor 
S.  P.  Thompson,  consists  in  the  insertion  of  inductive  shunt 
circuits  or  leaks  across  the  two  members  of  the  cable  or  between 
the  line  and  the  earth.  It  is  clear  from  the  explanations  already' 
given  that  the  distorsional  quality  of  the  line  depends  essentially 
upon  the  excess  of  numerical  value  of  the  product  CR  over  the 
product  LS  p^r  mile  of  line.  Hence,  since  CR  is  numerically 


TELEPHONY  AND  TELEPHONIC  CABLES   133 

larger  than  LS  for  any  ordinary  cable,  we  can  effect  the  adjust- 
ment either  by  increasing  L,  as  already  explained,  or  increasing 
the  insulation  conductance  S.  Thus  for  a  standard  telephone 
line,  where  R  =  88  ohms,  C  =  0'05  X  10~6  farad,  and  L  = 
O'OOl  henry,  we  should  have  to  reduce  the  insulation  resistance  to 
227  ohms  per  mile  to  bring  about  the  necessary  equalisation. 
This  might  be  done  by  putting  fifty  equidistant  shunts  per  mile, 
each  of  10,000  ohms,  between  the  members  of  the  cable. 

The  result,  however,  would  be  to  immensely  increase  the 
attenuation  constant  of  the  cable,  and,  although  it  would  equalise 
the  attenuation  for  different  frequencies  and  therefore  contribute 
to  produce  clearness  of  articulation,  it  would  certainly  decrease 
the  volume  or  loudness  of  the  sound,  and  loudness  is  as  essential 
as  clearness  for  intelligibility.  Even  if  we  did  not  lower  the 
insulation  to  the  full  amount  above  given,  yet  the  insertion  of 
suitable  non-inductive  shunts  across  the  cable  does  something  to 
assist  telephonic  transmission. 

Nevertheless  it  remains  evident  that  the  increase  of  leakage  in 
some  degree  acts  as  an  alternative  method  for  curing  distorsion 
in  the  case  of  telephone  cables. 

The  subject  of  the  effect  of  leakage  in  telephone  and  telegraph 
lines  is  complicated  by  the  nature  of  the  receiver  used.  The 
reader  will,  however,  find  some  valuable  information  on  this 
subject  in  Mr.  Oliver  Heaviside's  book  "  Electromagnetic 
Theory,"  Vol.  L,  §  213,  under  the  heading  of  "  A  Short  History 
of  Leakage  Effects  on  a  Cable  Circuit,"  in  which  the  effect  of 
leakage  on  signalling  speed  for  different  types  of  receiving 
instrument  is  most  clearly  explained. 

1O.  The  Theory   of  the  Thompson  Cable.— The 

theory  of  the  type  of  cable  suggested  in  1891  and  1893  by 
Professor  S.  P.  Thompson  for  overcoming  distorsion  has  been 
discussed  by  Dr.  E.  F.  Kosher  in  an  able  paper  following  the 
same  lines  as  the  discussion  of  the  Pupin  cable  already  given.1 

The  Thompson  cable  consists  of  a  lead  and  return  conductor 
between  which  at  equal  intervals  are  connected  shunt  circuits 

1  See  The  Electrical  World  and  Kinjineer  of  New  York,  Vol.  XXXVII.,  pp.  440, 
477,  and  5!0,  March  16tb,  23rd,  and  30th,  1901. 


134        PROPAGATION   OF    ELECTRIC   CURRENTS 

having  inductance  and  resistance  (see  Fig.  7).  The  problem 
to  be  discussed  is  the  right  distance  to  place  these  shunts  and 
the  value  of  their  impedance  so  as  to  effect  an  improvement  in  the 
distorsional  qualities  of  the  non-shunted  cable. 

Let  the  inductive  shunts  each  have  resistance  E0  and  induct- 
ance L0,  and  let  n  such  shunts  be  bridged  across  in  the  run  of 
the  cable.  Let  I  be  the  distance  between  the  transmitter  and 
receiver.  Let  the  cable  itself  have  resistance  11,  inductance  L, 
and  capacity  C  per  unit  of  length,  and  suppose  a  simple  harmonic 


FIG.  7.  —  Thompson  Cable  with  Inductive  Shunts. 

electromotive  force  denoted  by  the  real  part  of  Etjpi  be  operative 
'in  the  transmitter. 

Let  #0  +  jp  L0  =  z0  and  R  -f-  jp  L  =  z. 

Let  im  be  the  current  in  the  line  at  a  point  between  the  wth 
and  (??£  +  l)th  shunt  at  a  distance  x  from  the  with  shunt. 

Then  at  that  point  we  can  write  a  differential  equation  for 
the  current  tmas  already  proved  for  a  uniform  line,  viz., 


As  already  proved,  this  differential  equation  has  a  solution 
applicable  in  the  present  case  in  the  form 

im  =  Ki  Cos  fjiX  +  Kz  Sm.fjLX        .         .         .     (72) 
where  ^  =  -  C  (  -p*L  +jpR)  . 

If  JJL  =  /3  +  ja,  then,  as  already  shown, 


«=x/J 


.     (73) 


The  integral  (72)  expressing  the  value  of  inl.has  to  fulfil  n 
boundary  conditions  at  the  terminations  of  tbe  shunt  coils. 


TELEPHONY  AND  TELEPHONIC  CABLES   135 

Let  r/i,  r/2,  #3,  etc.,  be  the  currents  in  the  shunt  coils  ;  then 

9i  =  fa)x=un-(ii)*=Q,  etc.          .  .     (74) 

0m=(%-iXc=*/»-(0)*=o   •         •  •     (75) 

where  (io)x=ifn  stands  for  the  current  in  the  run  of  the  cable  in 
that  section  just  before  the  first  shunt  close  up  to  the  junction  of 
the  shunt  and  (ii)x=o  stands  for  the  current  in  the  section  after 
the  first  shunt  at  a  point  close  to  the  junction  of  the  shunt. 

Let  vi,  r2,  vs,  etc.,  be  the  potentials  atone  end  of  the  shunts, 
and  let  vi  ,  r2',  v3'  ,  be  the  potentials  at  the  other  ends.  Then 
ri  —  i'i  ,  etc.,  are  the  drops  in  potential  down  the  shunts. 

Let  Vm  stand  for  the  potential  in  the  run  of  the  cable  at  any 
point  between  the  mth  and  (w+l)th  shunt. 

Then  Vm  satisfies  a  differential  equation  of  the  type  of  (71), 
and  this  has  an  integral  like  (72),  viz., 

Vm=NlCosfjLX+Nz  Sin/x^       .         .         .     (76) 

also  (Vm)x  =  iin=vm  =  (Vm+1)x=0.        .        .        .     (77) 

using  the  same  notation  as  in  the  case  of  the  currents.    Likewise 

vm-vn;  =  E,gm+L,djf  .         .        .     (78) 

But  when  the  currents  and  potentials  are  steady  vm  —  vm' 
varies  as  A^pt. 

,    ....     (79) 


fJV  rli 

Now  it  is  clear  that  C=  -,  and  hence  from  (72)  and  (76) 


Therefore    vm=N»  and  vm+l=N,  Cos  ^+^2  Sin  &. 

tit  tii 

And  K!  =  —^  —  ^  ("«+«—  »w  Cos  p.  -  J 


V 

Therefore,  substituting  these  values  of  KI  and  7i2  in  (72),  we  have    . 

jpc 

Cos  fjiX  —  vm  Cos  /A  ( -  —  x 


136        PROPAGATION   OF   ELECTRIC   CURRENTS 

This  equation  is  correct  only  from  m  =  1  to  m  =  n  —  1,  but 
for  i0  and  in,  viz.,  the  currents  in  the  end  sections,  we  have  to 
develop  special  formulae.  It  is  not  difficult  to  see  that  the 
currents  in  the  transmitter  and  receiver  sections  are 


v,  Cos  fix  - 1  EeM  Cos  p  (±  -x\\       .     (81) 


in=  _  --^~-T  Cos  fi  (  L-x]  vn   .         .         .     (82) 
/*  Sin  |^  V2^       / 

We  can  now  write  the  boundary  equations. 

2/x  Sin  —  j 

Let  0-=-- --7r-1-  -4Sin2£-         .         .        .     (83) 


2M  Sin  , 

,+2=     -+200  .  .    (84) 


Then  the  boundary  equations  are  as  follows 


K?l"titL-o  (••  •  •  ^ 

(<T  +  2J    Vw_!  — Vw_2  — ^  =  0 

(cr+8)  Vn  —  Vn-! 

If    the    transmitting    and    receiving    instruments    have    no 
impedance,  then  //0  =  /«!  =  0,  7i  =  <r  =   --  4  Sin2  ^,  and  let 


Then  we  have 

I  ,       .  Sin  (2?z- 


as  an  equation  which  determines  the  potential  at  the  end  of  a 
shunt  coil. 

The  question  then  arises  how  far  apart  must  or  may  the 
inductive  shunts  be  placed  in  order  that  the  Thompson  cable  may 
be  electrically  equivalent  to  a  certain  uniform  line  called  the 
equivalent  conductor.  In  the  case  of  the  Pupin  loaded  line  the 
equivalent  conductor  is  a  conductor  having  the  same  total 
inductance  and  resistance  as  the  loaded  line,  but  spread 


TELEPHONY  AND  TELEPHONIC  CABLES   137 

uniformly,  and  we  have  shown  that  if  the  angular  distance 
between  the  coils  is  y  on  the  same  scale  that  the  wave  length 

is  2w,  then  as  long  as  ~  y  is  not  very  different  from  Sin  ~  y  a  line 

A  A 

so  loaded  may  for  transmission  purposes  be  replaced  by  the 
equivalent  uniform  line.  In  the  case  of  the  Thompson  line  we 
have,  however,  first  to  define  what  we  mean  by  the  "  corre- 
sponding uniform  conductor." 

Let  us  consider  the  equation  (83)  by  which  a-  is  determined, 
we  have 

2/x  Sin  p.  -  , 


If  jot  =  |3  +  ja  where  ft  is  large  compared  with  a,  then  the  wave 
length  on  the  unloaded  uniform  wire  is  A  =  27T//3,  and  the  angular 
distance  between  the  consecutive  coils  for  the  wave  length  A  is  y, 
where 

r-T—0       •  •      •   (88) 

r     n  A     »M 

If  then  y  is  so  small  that  Sin  y  =  y  nearly,  the  above  equation 
for  a-  can  be  written 


Hence  we  get  Sin  0  =  ^  —  »,,. 

2  0™ 

If  we  insert  in  the  above  equation  the  values  of  /x  and  ZQ,  viz., 
^0  =  PQ  +  j>L0  and  /z  =  \/  {  —  C  (—  y2L  -\-jpR)  \   we  reach  an 

equation, 

-^^C^p^+jpE,).  .     (90) 

in  which 


'     '     '  (92) 

Suppose  then  that  we  have  a  uniform  line  the  inductance  and 
resistance  of  which  per  unit  of  length  are  LI  and  RI  as  given  by 
the  above  equations,  its  capacity  per  unit  of  length  being  (7, 


138        PROPAGATION   OF   ELECTRIC   CURRENTS 

then  this  line  is  the  "  corresponding  uniform  line  "  with  which 
the  Thompson  cable  has  to  be  compared. 

We  can  now  prove  the  equivalence  of  the  Thompson  loaded 
line  to  the  equivalent  uniform  line  defined  as  above. 

If  ^  —  /3i  +  jai   we   have   /3i  =  T     where   AI   is   the   wave 

Ai 

length  for  the  frequency  p/2-x  in  the  corresponding  uniform 
conductor  just  defined.  If  Aj  is  represented  as  an  angle  2-77,  then 
the  angular  distance  between  two  successive  shunts  is  yi,  such  that 


If  we  assume  ^yi  is  so  small  that  ^yi  =  Sin  ^yi  nearly,  and 

also   2^/3x   so   small  that   e'J"    =  1  +  y-ft,   we  get    6  =  2  n^i, 

and  our  equation  (86)  for  the  value  of  vm  on  the  Thompson 
line  becomes  identical  with  the  value  for  a  corresponding 
uniform  cable  as  above  defined. 

Accordingly  we  can  summarise  the  results  by  saying  that  — 
A  loaded  cable  of  the  Thompson  type  with  inductive  shunts 
at  equal  intervals  is  equivalent  to  its  corresponding  uniformly 
loaded  cable  characterised  by  inductance  and  resistance  per  unit 
of  length  as  defined  in  equations  (91)  and  (92)  as  long  as  the 
sine  of  half  the  angle  denoting  distance  between  two  consecutive 
shunts  is  not  sensibly  different  from  the  angle  itself,  the  angle 
being  reckoned  on  such  a  scale  that  the  wave  length  for  the 
frequency  considered  is  equal  to  27r.  We  see  then  that  the  rule 
for  spacing  the  shunts  in  a  Thompson  cable  is  verbally  the  same 
as  the  rule  for  spacing  the  inductance  coils  in  a  Pupin  cable. 

The  difference  between  the  Pupin  and  Thompson  methods  is, 
however,  that  in  the  former  we  increase  the  effective  inductance 
of  the  cable  to  cure  distorsion  and  necessarily  increase  its  resist- 
ance as  well,  which  resistance  increase  we  must,  however,  keep 
as  small  as  possible.  In  the  latter  we  reduce  the  resistance  of 
the  cable  and  necessarily  reduce  its  effective  inductance  as  well. 
This  reduction  in  inductance  must,  however,  be  kept  as  small  as 
possible.  Hence  the  necessity  for  the  use  of  inductive  shunts 
and  not  inductionless  shunts, 


TELEPHONY  AND  TELEPHONIC  CABLES   139 

We  can  obtain  an  expression  for  the  average  attenuation  of 
the  Thompson  loaded  line  very  much  on  the  same  principles  that 
we  have  obtained  one  for  the  Pupin  line  in  §  8.  We  can 
consider  the  Thompson  line  to  be  made  up  of  a  series  of 
sections,  each  of  which  consists  of  a  double  length  d  of  plain 
line  having  a  propagation  constant  P  and  a  coil  connected  across 
the  end  having  an  impedance  X,.. 

Let  us  suppose  that  the  P.D.'s  across  the  ends  of  these  inductive 
shunts  are  denoted  by  Fi,  F2,  F3,  etc.,  then  each  section  may  be 
regarded  as  a  short  line  of  length  d  having  a  receiving  instrument 
of  impedance  Zr  across  its  far  end  and  a  P.D.  across  this  coil 
represented  by  Fn+1>  whilst  the  P.D.  across  the  sending  end  is 
VH.  Then  from  the  expressions  given  in  Chapter  III.,  if  FI  is 
the  sending  end  P.D.  and  Ii  the  sending  end  current  and  Zi 
the  final  sending  end  impedance  and  F2,  J2  and  Z2  the  corre- 
sponding quantities  for  the  receiving  end,  we  have 


Ia     Zi       ,  F2     Zr 
Hence  T  =  ^r  and  ^—  -&-  • 

-LI      ^2  "\      ^2 

Again,  since  the  sending  end  voltage  for  the  second  section  is 
equal  to  the  P.D.  at  the  ends  of  the  shunt  coil  terminating  the 
first  section,  we  have  for  the  second  section 


In  the  same  way  we  can  prove  that 


But  V1 

£= 

Hence  IT 

or  7  =  f^,.)»-2         ....    (95) 

J-\        £><i 

But  -"  =  t-1"11'1    where  P1  is  the  average  propagation  constant 

*i 
of  the  Thompson  line. 

Again  by  equations  (61)  and  (62)  in  Chapter  III. 

^1        _  ^0  _  /Qg\ 

Z~Z0  Cosh  Pd  +  Zr  Sinh  Pd 


140        PBOPAGATION   OF   ELECTBIC  CURKENTS 


We  have  then 


c-P'nd_ 


(97) 


Z0  Cosh  Pd+Zr  Sinh  Pd 
If  then  we  are  given  Z0,  Zr,  P,  n  and  d,  we  can  calculate 

€-p'nrf  =  Cosh  p'^-Sinh  P'nd. 

If,  therefore,  we  denote  by  a'  the  equivalent  attenuation  constant 
of  the  Thompson  line,  we  can  say  that  ^-a'nd  is  equal  to  the 
real  part  of  the  expression  on  the  right-hand  side  of  equation  (97), 
and  therefore  that  —a'nd  is  equal  to  its  Napierian  logarithm. 
We  can  then  find  af  in  terms  of  the  given  quantities. 
The  arithmetic,  however,  would  be  tedious. 
The  general  result  of  experimental  investigation  on  the  matter 
as  far  as  it  has  gone  goes,  however,  to  show  that  for  a  given 
amount  of  iron  and  copper  in  the  form  of  impedance  coils  it 
results  in  a  less  attenuation  constant  to  employ  them  in  the 
Pupin  fashion  as  coils  in  series  rather  than  in  the  Thompson 
fashion  as  coils  in  parallel. 

11.  Other  proposed  Methods  of  constructing 
Distorsionless  Cables. — In  addition  to  the  methods  com- 
prising the  addition  of  inductance  in  series  with  the  line  and  that 


FIG.  8. — Thompson  Transformer  Cable. 

of  inserting  inductive  shunts  across  the  line,  a  third  method  was 
proposed  by  Professor  S.  P.  Thompson  in  his  paper  on  Ocean 
Telephony  in  1893,  consisting  in  cutting  up  the  cable  into 
sections  inductively  connected  by  tranformers  (see  Fig.  8). 

This  plan  was  also  proposed  by  Mr.  C.  J.  Reed  in  1893,1 
although  it  had  been  previously  mentioned  and  specified  by 
Professor  S.  P.  Thompson. 

If  these  transformers  have  a  1  :  1  ratio  of  transformation,  or 

1  See  United  States  Patent  Specification  of  C.  J.  Reed,  Nos.  510,612  and  510,613. 


u 

u     j 

0 

i 

«* 

i 

£>                                    C 

8 

* 

5 

Qi 

^                            c 

?i 

n        i 

TELEPHONY  AND  TELEPHONIC  CABLES   141 

indeed  any  other  ratio,  they  are  electrically  equivalent  to  the 
addition  of  inductance  in  series  with  the  line  associated  with 
inductive  shunts  across  the  line.  Accordingly  it  has  been  proved 
mathematically  by  Dr.  E.  F.  Eoeber  that  such  a  transformer 
cable  as  in  Fig.  8  is  electrically  equivalent  to  the  arrange- 
ment shown  in  Fig.  9.1  He  has  also  proved  mathematically  by 
an  analysis  on  the  lines  of  that  already  given  for  the  Pupin  and 
the  Thompson  cable  that  the  transformer  cable  can  be  replaced 
by  a  certain  line  having  a  uniform  distribution  of  inductance, 
resistance  and  capacity  called  the  "  corresponding  uniform  line  " 
provided  that  the  intervals  between  the  transformers  are  short 


FIG.  9. 

compared  with  the  wave  length,  or  if  that  interval  is  denoted  by 
an  angle  y  on  the  same  scale  that  the  wave  length  is  denoted  by 
277,  then  the  transformer  line  differs  from  the  "  corresponding 

uniform  line  "  to  the  same  extent  that  Sin  ^y  differs  from  -^y. 

It  is  hardly  necessary  to  give  the  full  analytical  theory  of  this 
transformer  cable,  as  the  writer  is  not  aware  that  it  has  yet  been 
employed  in  practice,  but  the  reader  can  be  referred  to  Dr.  Roeber's 
article  for  additional  information. 

The  type  of  loaded  cable  suggested  by  Pupin  has,  however, 
come  into  extensive  use,  and  in  a  later  chapter  we  shall  describe 
some  of  the  results  of  practical  experience  and  the  confirmation 
they  give  of  the  above  theory. 

1  See  Wie  Electrical  World  and  Engineer  of  New  York,  Vol.  XXXVII.,  p.  510, 
1910.     Dr.  Roeber  calls  this  transformer  line  a  Reed-cable. 


CHAPTEE    V 

THE  PROPAGATION  OF  CURRENTS  IN  SUBMARINE  CABLES 

1  .  The    Differential    Equation    expressing  the 
Propagation  of  an  Electric  Current  in  a  Cable.— 

If  we  assume  a  cable  to  have  resistance  R,  inductance  L, 
capacity  C,  and  leakance  S,  all  per  unit  of  length,  and  if  the 
current  at  any  distance  x  from  the  origin  at  any  time  t  is  i  and 
the  potential  is  v,  then  we  have  seen  (see  Chapter  III.)  that  we 
can  express  the  state  of  affairs  at  that  point  x  by  two  differential 
equations,  viz., 


' 


The  first  of  these  equations  expresses  the  fact  that  the  fall  in 
potential  down  an  element  of  the  cable  is  due  to  the  combined 
effect  of  resistance  and  reactance  or  inductance,  and  the  second 
that  the  change  in  the  value  of  the  current  in  passing  along  an 
element  of  the  cable  is  due  to  the  combined  effect  of  capacity  and 
of  leakage.  If  we  differentiate  the  first  equation  with  regard  to 

cl^i 
x  and  the  second  with  regard  to  t  and  eliminate  ,     ,.  we  obtain 

0  .     (2) 


and  a  similar  equation  in  i  can  also  be  reached  by  reversing  the 
order  of  the  differentiations.  The  above  differential  equation  (2) 
is  of  the  type 


The  full  discussion  of  this  equation  would  lead  us  into  mathe- 
matical questions  of  an  advanced  nature.     Suffice  it  to  say  that 


CURRENTS   IN   SUBMARINE   CABLES  143 

it  can  be  satisfied  by  many   functions  of  x  and  t.     Thus  for 

instance  it  can  be  satisfied  by  a  function  of  the  form 

y  —  €~rt*  Sin  bx,  provided  there  are  certain  relations  between  the 

constants. 

Thus  if  v  =  €~at  Sin  bx,  and  we  find  the  values  of  -^    -^ 
and  T~I  from  the  above  expression  and  substitute  them  in  (2), 

we  have 

CLa'*-(BC+LS)a+BS  +  b2  =  0    .         .         .     (4) 

Solving  the  above  quadratic  equation  we  obtain 


IB     Sy      I 

iU~cy  ~~CL 

The  quantity  b  is  determined  by  the  distribution  of  potential 
along  the  origin  of  time  or  when  t  =  0.  If  then  we  take  a  point 
at  a  unit  of  distance  from  the  origin  or  take  x  =  1,  we  have 
v  —  Sin  b  or  b  =  Sin"1  r.  In  other  words,  b  is  the  inverse 
sine  of  the  potential  at  a  unit  of  distance  from  the  sending  end 
at  the  instant  from  which  time  is  reckoned. 

Suppose  we  assume  an  initial  distribution  such  that  the 
potential  varies  along  the  cable  according  to  a  simple  sine  law  of 
distribution.  Then  St/h  is  the  wave  length.  If  then  the  con- 

stants  of   the  cable   are   such   that  T(T~(^\    ig  greater  than 

b'2 

Y^  the  quantity  under  the  square  root  sign  in  (5)  is  real,  and 

the  quantity  a  is  therefore  real,  and  the  potential  at  any  point  in 
the  cable  dies  away  exponentially  or  according  to  a  geometric 

1    /  7?      S!\  2 

law  of  decrease,  but  without  oscillations.    If,  however,  ^  \jj~(j) 

b2 
is  less  than    —  ;  the  value  of  a  is  a  complex  quantity,  viz., 


**      ....     (6) 

TO          ~t      /  T~) 

where  (f  stands  for         ~LC~±  \Z~ 


Hence  0=«  Tvi^c/1  Sin  bx  (Cos  qt—  j  Sin  qt), 

which  indicates  that  there  is  at  any  fixed  point  in  the  cable 


144        PROPAGATION   OF   ELECTRIC   CURRENTS 

a   decadent   oscillation   of   potential   with   time,    the    potential 
ultimately  becoming  zero. 

Another  solution  of  the  differential  equation  (2)  more  applic- 
able in  the  case  with  which  we  are  concerned  is 

v  =  A*-\  (l+?)'  Sin  (bx  ±qt)          .         .         .     (7) 
This  represents  a  damped  or   decaying   oscillation   of   wave 
length  2  ir/b  propagated  with  a  velocity  q/b  along  the  cable. 

It,     S 

If  the  constants  of  the  cable  have  such  relation  that  -=-  —  -=0, 

Li      0 

that   is  if   CR  =  LS,    or   if   the   cable  is  distorsionless,  then 
the  quantity  a  is  always  real  and  q*  =  j^,  or  ^  =     /,-—  ,  that 

is,  .  the  oscillations  of  all  frequencies  are  propagated  with  the 
same  velocity,  1/VLC. 

If  we  assume  that  v  is  a  simple  periodic  quantity  and  can  be 

represented    by   the    real    part    of    Atjpt,   then  -r,  =  jpv    and 

£p  =  —  p2v  ,  so  that  the  differential  equation  (2)  then  takes  the 
form 


or  =(S+jpC)(It+jpL)v  .  •        •     (8) 

This  is  the  equation  we  have  already  fully  discussed  in  dealing 
with  the  propagation  of  currents  in  telephone  cables  where  we 
can  assume  that  v  varies  in  accordance  with  some  function  of 
the  time  which  by  Fourier's  theorem  can  be  resolved  into  the 
sum  of  a  number  of  simple  periodic  terms. 

In  dealing  with  the  problem  of  the  submarine  telegraph  cable, 
however,  the  differential  equation  can  be  somewhat  simplified  as 
in  the  next  section. 

2.  The   Discussion  of  the  Telegraph  Equation. 

—  In  telegraphic  signalling  the  changes  of  current  or  potential  at 
the  sending  end  are  generally  so  slow  and  the  inductance  of  the 
cable  so  small  that  the  quantity  pL  or  27mL,  where  n  is  the 
frequency,  is  small  compared  with  the  resistance  R.  Also  the 


CURRENTS   IN    SUBMARINE    CABLES  145 

leakage  is  so  small  that  S  is  negligible.     Hence  the  general 
equation  (2)  reduces  to 

£=*'*     .....     (9) 

This  equation  is  called  the  "  telegraph  equation."  It  first 
presented  itself  in  connection  with  a  problem  on  the  conduction 
of  heat  in  a  bar,  but  was  established  as  the  fundamental 
differential  equation  in  the  theory  of  the  telegraphic  cable 
by  Lord  Kelvin  (then  Professor  William  Thomson)  in  a 
celebrated  classical  paper  ''On  the  Theory  of  the  Electric 
Telegraph  "  communicated  to  the  Royal  Society  of  London  in 
May,  1855  (see  "  Mathematical  and  Physical  Papers  of  Lord 
Kelvin,"  Vol.  II.,  article  Ixxiii.,  p.  61). 

The  discussion  of  this  equation  as  given  by  Lord  Kelvin  is 
not  exactly  suited  for  an  elementary  treatise,  but  it  has  been 
simplified,  especially  by  the  late  Professor  Everett  in  a  volume 
on  electricity  and  magnetism  forming  part  of  a  revised  edition  of 
Deschanel's  "  Natural  Philosophy."  We  shall  follow  the  general 
method  of  this  latter  treatment. 
Consider  the  equation 

d*v     -dv 


The  following  are  two  particular  solutions  :— 

v  =  B+Dx        .....     (11) 
v=A€-*?'Smpx     ....     (12) 
where  k  =  1/RC  and  A,  B,  and  C  are  constants. 

It  is  clear  that  (11)  satisfies  (10).  Also,  if  (12)  is  differen- 
tiated twice  with  regard  to  x  it  gives  —  /3*v,  and  if  differentiated 
with  regard  to  t  and  multiplied  by  EC  =  l/k  we  have  also 
—  /32r.  Therefore  (12)  is  a  solution  of  (10)  subject  to  k  = 
l/R  C.  A  precisely  similar  equation  to  (10)  presents  itself  in 
considering  the  conduction  of  heat  along  a  bar  and  also  the 
diffusion  of  salt  through  a  tube  of  water  or  other  solvent. 

Thus  if  we  have  a  metal  bar  of  unit  cross  section  and  thermal 
conductivity  k,  composed  of  a  material  of  specific  heat  c,  and  if 
we  consider  a  small  section  of  length  Sx,  and  if  the  temperature 

on  one  side  of  the  section  is  v  and  on  the  other  v  +  ^  8x, 
E.C.  L 


146   PEOPAGATION  OF  ELECTRIC  CURRENTS 


s/jj 

the  temperature  gradient  down  the  section  is  j-   and  the  rate 

dv 
of  flow  of   heat   into  the  section  is   k  -j-  .     Hence  the  rate  of 

accumulation  of  heat  in  the  section  is  expressed  by  j-  (&  y  )  &#• 


But  this   can   also  be   expressed   by  c£r  j.,  where  c&x   is  the 

amount  of  heat  required  to  raise  the  section  &x  one  degree  in 
temperature.  Equating  these  two  identical  expressions  we  have 

d2v     c  dv 

dx"2  =  k~di' 

Again,  if  we  have  a  tube  of  solvent  of  unit  section  and  con- 
sider the  diffusion  of  some  salt  along  it,  we  have  a  precisely 
similar  equation,  only  in  this  case  k  stands  for  the  diffusivity  of 
the  salt  and  c  for  the  mass  of  salt  required  to  produce  unit 
concentration  per  cubic  unit  of  volume  of  the  solvent.  Lastly, 
the  same  type  of  differential  equation  comes  to  notice  in  con- 
sidering the  gradual  penetration  of  an  electric  current  into  a 
conductor,  since  all  the  above  cases,  propagation  of  potential 
along  a  submarine  cable,  salt  diffusion,  and  thermal  conduction 
are  really  cases  of  diffusion  of  electricity,  matter,  or  heat. 

3.    The    Theory    of    the    Submarine     Cable. 

Suppose  a  cable  of  length  I  to  have  its  distant  or  receiving  end 
earthed  and  to  have  a  distribution  of  potential  made  along  it 
which  is  represented  by  the  equation 

mirX 

v  =  A  Sin  —T-          ....     (13) 

This  means  that  the  potential  at  the  sending  end  (x  =  0)  is  to 
be  zero,  and  that  at  the  receiving  end  (x  =  I)  is  to  be  zero,  and 
that  a  maximum  potential  v  =  A  exists  at  some  intermediate 
point. 

Let  this  potential  distribution  be  left  to  itself,  then  the  first 
question  is  what  function  of  the  distance  x  and  the  time  t  will 
represent  the  distribution  after  the  lapse  of  any   stated    time. 
It  must  be  such  a  function  that  it  satisfies  the  equation 
d?v_Tf~dv        d2v  _1  dv 

dtf-^dt  or  dx*~kdT 


CURRENTS  IN  SUBMARINE  CABLES     147 


Also   it    must    satisfy  the   boundary  conditions ;  that  is,  have  a 
zero  value  both  for  x  •=.  0  : 
/.  r=  0.     Such  a  function  is 


zero  value  both  for  x  =  0  and  x  =  I  and  a  value  A  Sin  -    -  for 


..     (14) 

For  it  obviously  reduces  to  (13)  when  t  =  0  and  it  is  zero  when 
x  =•  0  or  x  =  /.    If  twice  differentiated  with  regard  to  x  it  becomes 

— ^-  v,  and  if  differentiated  with  regard  to  t  it  yields  —  m*nv. 

2 

Hence  if  u  =  -ftj™  the  expression  (14)  satisfies  the  differential 

-Zl/  O  6^ 

equation  (10). 

Accordingly  it  is  seen  that  the  expression  for  the  distribution 
of  potential  at  zero  time,  viz., 

.    ~ .        tJlTT 

0=4  Sin -y- a;        ....     (15) 
is  changed  by  lapse  of  time  t  to  the  expression 

v  =  A  (€-«*'")  Sin7-^?    .         .         .         .     (16) 

9 

and  both  of  these  satisfy  all  the  conditions ;  provided  u  =  ^^ . 

If  we  assume  any  distribution  of  potential  it  must  be  capable 
of  being  represented  by  a  single  valued  curve,  because  the 
potential  can  only  have  one  value  at  any  one  point  at  the  same 
instant.  Now  such  a  curve  can  be  resolved  by  the  Fourier 
analysis  into  the  sum  of  a  number  of  simple  periodic  or  sine 
curves  of  different  amplitude  and  phase.  Hence  if  we  can 
express  in  the  form  of  a  Fourier  series  the  initial  distribution 
of  potential,  then  after  the  lapse  of  a  time  t  this  distribution  if 
left  to  subside  will  be  changed  into  one  which  is  expressed  by 
multiplying  each  term  of  the  above  Fourier  series,  which  is  a 

term    of    the    form   A  Sin  - -,— ,   by   an    exponential   factor   of 

the  form  e~ m2'lt,  since  each  term  of  the  original  and  each  term 
of  the  so  altered  series  satisfies  the  differential  equation  and  also 
the  boundary  conditions. 

9 

For  the  same  cable  the  quantity  u  =    J(       has  a   constant 

L    2 


148        PROPAGATION   OF   ELECTRIC   CURRENTS 

value,  and  hence  the  exponential  factors  for  the  different  terms 
will  have  the  same  value  at  times  t  which  are  inversely  as 
w2  or  directly  proportional  to  the  square  of  the  wave  length  A 

because   the   quantity  -y-  must   be  equal   to   ~.     Accordingly 

the  terms  representing  waves  of  short  \vave  length  die  away  more 
quickly  than  long  ones. 

Suppose  then  that  at  the  sending  end  of  the  cable  we  apply 
one  pole  of  a  battery  and  raise  the  end  to  a  potential  V,  the 
receiving  end  remaining  connected  to  earth.  There  will  after  a 
time  be  a  final  distribution  of  potential  gradually  diminishing 
from  V  at  the  sending  end  to  zero  at  the  receiving  end,  and  the 


FIG.  1. 

potential   at   any  distance   x  from   the   sending    end    will    be 
represented  by  the  expression 

v=V^ (17) 

For  this  expression  (17)  represents  a  potential  gradient  in  the 
form  of  a  straight  line.  (See  Fig.  1.) 

If  this  steady  state  is  altered  by  putting  the  sending  end  to 
earth  at  the  time  t  =  0,  then  the  potential  becomes  zero  at  the 
sending  end  or  v  =  0  for  x  =  0,  and  at  every  other  point  it  is 

represented  by  v  =  V  —j-  - 

To  find  the  subsequent  distribution  we  have  to  expand  the 
last  expression  into  a  series  of  sine  terms  and  find  the  co- 
efficients. 

I  —  X  TlX  _,.      %TfX   ,  t      ft'      mnX  . 

If    y  =  — T-  =  A1  Sin  -f-f^a  Sin  — j-  -fete.  -\-Am  Sin  — *—      .     (18) 

III  i 


CURRENTS   IN    SUBMARINE    CABLES  149 

We  proceed  to  find  the  values  of  the  co-efficients  AI,  A2,  .  .  .  Am 
in  the  manner  already  explained  in  Chapter  IV.     Multiply  both 

sides  of   the  expression  by  Sin  — ~  &x  and  take  the  average 

value   of   each  term   between  x   —  0  and  x  =   2/.      Then  all 
products  on  the  right  hand  side  vanish  except  one,  because  the 
average  value  of  such  an  expression  as  Sin  n  6  Sin  m  0  is  zero 
when  taken  over  one  complete  period. 
Hence  we  have  left 


Now         J  ^SnSp  &*= jSin  "^  to-f|  Sin"^  &r 

but  .  f  Sin  ^  &r=  -  —  Cos  '-^ 

J  t  m?r  I 

,  f  ?/l7T^  /2  .       77l7T^          Za? 

also  1  x  Bin  — r—  8^= 2— ^  Sm  — ^ 


^       m-rrX 

Cos  - 


Hence 


l  —  x        mi 
^—Sm— 

i  I 


mirx  .  /  mTra;        Z  WTT#  ,      a?      _     m>ra; 

8x=-    -Cos-y-    rrsSin-^  —  h-      -  Cos  —  ;— 

lllTT  I  m^TT2'  I  tllTT  I 

(l  —  x)  _     rmrx        I  m-n-x 

-  "*      VyOS^  7~"  n       &    Olll  -,   -    . 

tllTT  I  m27T2  I 

The  value  of  this  last  integral  between  the  limits  x  =  0  and 
x  =  2Z  is  -~1-. 


Again,  the  integral     sin^          &*=  Cos 


and  the  value  of  this  between  the  limits  x  =  0  and  x  =  2£  is  /. 
Hence  the  result  of  multiplying  both  sides  of  equation  (18) 

by  Sin  —  ^—  Sx  and  integrating  between  x  =  0  and  x  =  Zl  or 

taking  2£  times  the  average  value  of  each  term  is  to  give  us  the 
equation 


or  A=  — 


150        PROPAGATION   OF   ELECTRIC   CURRENTS 

I  —  x 
Hence  for  the  expansion  of  —7—  we  have 

I  —  X      2  (n.      TTX   .   1  „        27TX'   ,   1  „        STTX 


Therefore  the  potential  at  any  point  x  in  the  cable  at  zero  time 
or  when  t  =  0  is  expressed  by 

O  m=  <*>    /  1  7>?7rr\ 

v^V~    5    (4  Sin™)         .         .         .     (21) 

v  m  =  1  \w  {      / 

where    2    stands   for    the    sum    of   a   number    of    terms   like 

1  0.     m-nx  ,    .  .  . 

—  Sin  -~Y-  ,    m    being    given    various    values,   from   m  -  -  1    to 

m  •=  infinity. 

Each  of  these  terms  is  therefore  a  term  of  the  type  A  Sin  —  j—  . 

We  can  therefore  find  an  expression  for  the  potential  at  any 
point  in  the  cable  after  the  lapse  of  a  time  t  when  the  initial 
distribution  is  left  to  subside  by  simply  multiplying  each  sine 
term  of  the  above  series  by  a  factor  of  the  type  e~m2"',  as  already 
explained. 

If  then  we  denote  by  ?'0  the  potential  at  a  distance  x  at  a  time 
t  —  0,  and  by  vt  the  potential  at  x  after  a  time  t,  we  can  express 
r0  and  vt  as  follows  : 


(22) 

77  m=1 


(23) 
m 

Suppose  next  that  we  alter  the  origin  of  time,  and,  instead  of 
reckoning  the  origin  of  time  from  the  instant  when  the  sending 
end  is  earthed  after  having  been  raised  to  a  potential  V  and  kept 
there  long  enough  for  the  whole  potential  distribution  to  reach  a 
steady  state,  let  us  suppose  that  the  sending  end  has  a  battery 
applied  to  it  or  a  source  of  steady  potential  V,  and  that  we 
reckon  the  time  from  this  instant  of  applying  the  voltage  V  to 
the  sending  end.  At  that  instant  when  t  =  0,  the  potential  at 
the  sending  end  jumps  up  to  F,  and  at  all  other  points  rises  up 
gradually  to  a  limit  which  is  given  by  the  expression  (22). 

Hence  at  any  time  t  reckoned  from  the  instant  of  applying 
the  steady  voltage  to  the  sending  end,  the  potential  v  at  any 


CUKEENTS  IN  SUBMAKINE  CABLES 


151 


distance  x  from  that  sending  end  is  given  by  the  difference  between 
the  values  of  v0  and  vt,  as  given  in  (22)  and  (23).  In  other 
words,  if  we  apply  a  steady  potential  V  to  the  sending  end  at  a 
time  t  =  0,  then  at  a  time  t  and  at  a  distance  x  the  potential  in 
the  cable  is  given  by 


m=i 


m 


(24) 


The  part  of  the  expression  in  square  brackets  will  be  denoted  by 

<£  (x,  t),  so  that 

v  =  V<f>  (x,t)  .....    (25) 

gives  the  potential  at  any  time  and  place.     This  function  <£  (x,  t) 
satisfies  all  the  conditions.     It  satisfies  the  differential  equation 

T-2  =  11C  j~.  ,   for    it    is    the    difference    of     two    expressions 
which  separately  satisfy  it.     It  also  fulfils  the  boundary  con- 


FIG.  2. 


ditions,  because  when  t  =  0   0  (x,  t)  =  0,  and  when  t  =  infinity 


(x,  t)  = 


Hence    it   must    be   the   expression    for    the 


potential  in  the  cable  at  a  distance  x  and  at  a  time  t. 

We  may  represent  it  graphically  as  follows  :  —  Let  AB  (Fig.  2) 
represent  the  cable,  A  being  the  sending  end.  Let  a  voltage  V 
be  applied  at  the  sending  end,  represented  by  AC.  Then  at  a 
time  t,  after  the  application  of  this  voltage,  the  potential  all 
along  the  cable  will  be  represented  by  the  ordinates  of  the  firm 
line  curve  CDB.  After  a  long  time  this  potential  everywhere 
approximates  to  a  uniform  fall  represented  by  the  ordinates  of 
the  dotted  line  CB.  The  ordinate  of  the  firm  line  curve  corre- 
sponding to  any  distance  x  represents  the  potential  v  and  is  given 


152        PROPAGATION   OF   ELECTRIC   CURRENTS 

by  the  expression  v  =  F<£  (x,  t).  The  current  i  in  the  cable  at 
any  point  is  obtained  from  the  potential  v  by  differentiation  with 
regard  to  x,  since  by  Ohm's  law 


Hence,  performing  the  operation  denoted  by  (26)  on  v  =  V  $  (x,  t), 
we  obtain  the  expression  for  the  current  i  at  any  time  t  and  any 
distance  x,  viz., 


The  current  at  the  receiving  end  will  be  denoted  by  /,,,  and  it  is 
obtained  from  (27)  by  putting  x  =  I  and  giving  m  increasing 
integer  values  from  1  to  oo.  Hence 


It  is  convenient  to  denote  e~M*  by  6  and  to  write  (28)  in  the 
form 


Ir=r  2       -e+ei-0»+ei*-e^+e™-zte  .     (29) 

Ml       \j&  ) 

The  above  is  the  expression  for  the  current  flowing  into  the 
earth  at  the  receiving  end  at  any  time  t  after  applying  a  steady 
voltage  V  at  the  sending  end.  Since  0  is  a  proper  fraction,  the 
series  in  the  brackets  in  (29)  is  rapidly  convergent,  and  in 
general  it  is  quite  sufficient  to  take  the  sum  of  the  first  six  or 
seven  terms  to  obtain  a  close  approximation  to  the  actual  value. 

If  we  are  given  the  numerical  value  of  the  whole  resistance  of 
the  cable  in  ohms,  which  is  equal  to  111,  where  I  is  the  length, 
and  the  whole  capacity  of  the  cable  in  farads,  which  is  equal 

2  9f87 

to  Cl,  then  we  can  at  once  calculate  u  =  7]J772  —    ci-IU*     an^ 

hence  we  can  calculate  e~nt  =  0  from  the  expression 

0  =  e-««  =  Cosh  ut  -  Sinh  ut 

for  any  assigned  value  of  the  time  t.  We  can  then  find  6*,  09, 
etc.,  easily  by  the  use  of  a  slide  rule  or  table  of  logarithms.  For 
Iogi004  =  4  Iogi00,  and  therefore  04  =  logic"1  (4  logic  #),  etc.  It 
is  most  convenient  to  arrange  the  series  as  follows  : 


CURRENTS   IN    SUBMARINE    CABLES  153 

We  shall  denote  the  above   series  by  f(u,  t).     Accordingly  we 
have  for  the  received  current 

OT7" 

Jr=m/(M)  .     (30) 

and  for  any  assigned  value  of  the  time  t  we  can  calculate  the 
current  Ir  flowing  to  earth  at  the  receiving  end. 

4.  Curves  of  Arrival.  —  The  series  denoted  by  /  (u,  t)  has 

the  curious  property  that  its  value  is  zero  for  all  values  of  t 
from  t  =  0  up  to  *  =  CRP  X  0*0233  nearly. 
Consider  the  series 

0-04+/?s>-01G+025-<93G,  etc. 

Assume  t  =  0  ;  then  0  =  e~tlt  =  1,  and  the  series  (28)  becomes 
equal  to  1-1  +  1-1  +  1-1  +  1,  etc.,  to  infinity.  Let 
the  sum  of  this  last  series  to  infinity  be  denoted  by  S  ;  then 

5  =  1-1  +  1-1  +  1-1  +  1,  etc. 
Hence  5-1=  -1  +  1-1  +  1-1  +  1-1,  etc. 

Adding  the  above  two  series,  we  have 

25-1  =  0  or  S  =  l. 

Accordingly  the  sum  1  —  1  +  1-1  +  1,  etc.,  to  infinity  is 
equal  to  ,  and  therefore  the  series 


f(u,  t)=     -0+04  -00+  0io  _025  +  086,  etc., 

is  equal  to  zero  when  0  =  1. 

Also  it  can  be  shown  by  trial  that  for  any  value  of  6  between 
0—1  and  0  =  0'8  or  0*9  the  value  of  /(//.,  t)  is  zero. 

Thus  if  0  =  0-79  we  can  easily  find  that  04  =  0'389,  09  =  0119, 
6>16  =  0-023,  and  025  =  0'003. 

Hence  0  +  09  +  025  =  0-912  and  <94  +  01G  =  0-412.     Therefore 


and/(if,  0=0  when  0  =  e""'  =  0'79.  Also  it  can  be  shown 
that  if  0  =  0-9,  then  0  +  09  +  025  =  1-38,  and  04  +  016  =  '88, 
and  therefore  f  (n,  t)  =.  0. 

Lord  Kelvin  originally  gave  6  =  0*75  as  the  limiting  value 


154        PROPAGATION   OF   ELECTRIC   CURRENTS 

required  to  make  /  (M,  t)  equal  to  zero,  and  he  denoted  the  time 
corresponding  to  this  by  the  letter  a.1 

Since  6  =  €~ut,  we  have  t  =  -  Logcfgl,  and  if  6  =  0'75  then 

1          /4\ 
t=  -l°ge(g).      Hence    Lord    Kelvin's    symbol   a   is    a    time 

U  \6/ 

such  that 


Professor  Fleeming  Jenkin,   another  great  telegraphic  autho- 
rity, gave  as  the  limiting  value  0  =  0*79  =  10'0'1. 


Tirrue  reckoned,  frorrv  instCLrtt  of  depressing  Sending  Key. 

FIG.  3.  —  Curve  of  Arrival. 

Now  loge  (100'1)  =  0-23,  and  7i2  =  9'87. 
Accordingly  we  can  say  that 

O-23  =  CR  x  0-0233    .        .        .     (31) 


where  C  and   R  denote  the  capacity  in  farads  and  resistance 
in  ohms  of  the  whole  cable. 

Hence  if  the  key  is  put  down  at  the  sending  end   connecting 
that  end  with  a  battery  of  constant  potential  V,  then  during  an 

1  See  Lord  Kelvin,  "On  the  Theory  of  the  Electric  Telegraph,"  Proc.  Roy.  Sue., 
London,  May,  1855,  or  "Mathematical  and  Physical  Papers,"  Vol.  II.,  p.  71. 


CURRENTS  IN  SUBMARINE  CABLES 


155 


interval  of  time  equal  to  a  defined  as  above,  no  current  capable 
of  being  detected  by  any  receiving  instrument,  however  sensitive, 
would  be  found  flowing  to  earth  at  the  receiving  end.  If, 
however,  the  sending  key  is  kept  down,  then  the  current  will 
begin  to  rise  at  the  receiving  end  and  steadily  increase.  After 
an  interval  equal  to  about  4a  it  will  reach  nearly  half  its  final 
value,  and  after  an  interval  Wa  it  will  reach  a  final  steady 
value. 

If  we  plot  a  curve    the  ordinates  of  which  denote   to   some 
scale  the  received  current  and  the  abscissae  the  time  reckoned 


0-5 


0-4- 


03 


2345          67 
vut 

FIG.  4. — Curve  of  Arrival. 


JO 


from  the  instant  of  applying  the  battery  at  the  sending  end, 
the  curve  so  drawn  is  called  a  curve  of  arrival.  It  is  generally 
drawn  with  abscissas  representing  ut  and  ordinates  representing 
/  (11,  t),  and  has  the  form  represented  in  Fig.  3. 

Lord  Kelvin  was  the  first  to  give  in  1855  curves  of  arrival 
drawn  for  different  conditions. 

The  table  below  gives  values  of  /  (n,  t)  for  various  values  of 
•ut  calculated  by  Professor  J.  D.  Everett,  and  the  curve  in  Fig.  4 
graphically  represents  these  values. 


156        PEOPAGATION   OF   ELECTEIC   CUEBENTS 


The  value  of  /(//,  t)  approximates  to  0'5  as  ut  reaches  a  value 
of  about  10  and  upwards.     Below  u  =  0'23  f(u,  t)  =  0. 


ut. 

/(*,  0- 

ut. 

/(*,  0- 

ut. 

/(»,  0- 

0-1 

•000 

1-5 

•279 

2-9 

•445 

0-2 

•000 

1-6 

•300 

3-0 

•450 

0-3 

•001 

1-7 

•318 

3-1 

•455 

0-4 

•006 

1-8 

•335 

3-2 

•459 

0-5 

•018 

1-9 

•350 

3-3 

•463 

0-6 

•037 

2-0 

•365 

3-4 

•467 

0-7 

•062 

2-1 

•378 

3-5 

•470 

0-8 

•091 

2-2 

•389 

3-6 

•473 

0-9 

•121 

2-3 

•400 

3-7 

•475 

1-0 

•150 

2-4 

•409 

3-8 

•478 

1-1 

•179 

2-5 

•418 

3-9 

•480 

1-2 

•207 

2-6 

•426 

4-0 

•482 

1-8 

•233 

2-7 

•433 

5 

•493 

1-4 

•257 

2-8 

•439 

10 

•500 

The  interval  of  time  approximately  equal  to  0'0233  multiplied 
by  the  product  of  the  total  resistance  of  the  cable  in  ohms  and 
its  total  capacity  in  farads  is  called  the  "  silent  interval,"  and, 
no  matter  what  the  voltage  applied  at  the  sending  end,  no 
measurable  current  will  flow  out  at  the  receiving  end  to  earth 
until  after  the  lapse  of  this  time. 

After  a  time  about  ten  times  the  silent  interval  has  elapsed 
the  current  at  the  receiving  end  will  have  reached  its  full 
possible  value.  The  possible  speed  of  signalling  is  therefore 
closely  connected  with  the  duration  of  the  silent  interval. 
Since  the  silent  interval  a  varies  inversely  as  the  value  of  u  for 
the  cable  and  as  u  varies  inversely  as  the  product  CliP  or  the 
product  of  the  total  resistance  and  total  capacity,  we  can  say 
that  cables  have  equal  sending  power  for  which  the  value  of 
CRLZ  is  the  same. 

For  any  given  type  of  receiving  instrument  the  apparent  time 
occupied  in  the  transmission  of  a  signal  varies  as  the  square  of 
the  length  of  the  cable  for  cables  of  equal  capacity  and  resistance 
per  unit  of  length.  The  curve  of  arrival  can  be  actually  drawn 
by  such  a  receiving  instrument  as  the  syphon  recorder. 


CURRENTS  IN  SUBMARINE  CABLES 


157 


5.  The  Transmission  of  Telegraphic  Signals 
along  a  Cable. — We  have  next  to  consider  the  mode  of 
making,  and  the  effect  of  transmission  along  the  cable  on  tele- 
graphic signals. 

The  alphabetic  code  usually  employed  in  cable  telegraphy 
is  the  International  Morse  Alphabet,  according  to  which  each 


FIG.  5. — Syphon  Kecorder  for  Submarine  Cable  working  as  made  by 
H.  Tinsley  &  Co. 

letter  of  the  alphabet  is  denoted  by  one  or  more  intermittent 
applications  of  a  constant  potential  battery  to  the  sending  end  of 
the  cable,  such  application  being  made  by  a  key  which  connects 
the  cable  to  the  battery  for  a  certain  short  interval  of  time. 

The  battery  of  voltaic  cells  used  has  its  centre  connected  to  the 
earth,  and  a  key  is  employed  which  connects  either  one  or  other 
terminal  of  the  battery  to  the  sending  end  of  the  cable  and  there- 
fore raises  it  either  to  a  positive  potential  +  J7  or  lowers  it  to  a 
negative  potential  —  V. 


158       PROPAGATION* OF  ELECTRIC   CURRENTS 


In  signalling  over  land  lines  by  hand-made  signals  the  alpha- 
betic signals  are  composed  of  short  and  long  signals  called 
respectively  a  dot  and  a  dash. 

Thus  the  letter  A  is  represented  by  a  dot  followed  by  a  dash 


Tinue 


0 


Dot,  Signal 


FIG.  6. 


(  -  —  ).  The  dot  is  made  by  connecting  the  sending  end  of  the 
line  for  a  short  interval  of  time  with  one  terminal  of  a  battery. 
This  is  then  removed  and  after  an  equal  space  of  time  connected 
again  for  a  period  about  three  times  as  long  to  form  the  dash. 


O 


T    Time  cu&is. 


FIG.  7. 

The  currents  into  line  are  thus  always  in  the  same  direction,  but 
vary  in  duration. 

In  the  case  of  cable  signalling  the  currents  which  form  the  dot 
and  dash  signals  are  always  of  the  same  duration,  but  differ  in 
sign  or  direction,  those  forming  the  dashes  being  say  positive 
currents  and  those  forming  the  dots  being  negative  currents. 
The  receiving  instruments  are  therefore  differently  constructed. 


CURRENTS   IN    SUBMARINE   CABLES  159 

For  the  land  line  hand  sending  either  a  needle  instrument  or 
else  a  Morse  Inker  is  employed  when  printed  signals  are  required, 
and  the  message  is  printed  down  in  dots  and  dashes  on  paper 
strip. 

In  the  case  of  submarine  cables  the  receiving  instrument  used 
is  the  syphon  recorder  in  which  a  delicate  pen  moves  over  a 
strip  of  paper,  and  the  dot  and  dash  signals  are  made  by  slight 
but  sudden  deflections  to  the  right  or  left  (see  Fig.  5). 

To  make  a  dot  signal  the  positive  battery  pole  is  applied  to 
the  sending  end  of  the  cable  and  causes  the  potential  there  to 
rise  suddenly  to  +  J7«  After  an  interval  of  time  T  the  battery 
is  removed  and  the  end  put  to  earth.  The  variation  of  potential 
at  the  sending  end  may  therefore  be  represented  by  the  line  in 
Fig.  6. 

To  make  a  dash  signal  the  same  process  is  followed  with  the 
reversal  of  the  battery  pole,  so  that  the  variation  of  potential  at 
the  sending  end  in  making  the  dash  signal  is  represented  by  the 
firm  line  in  Fig.  7. 

We  have  then  to  consider  the  nature  of  the  potential  changes 
at  distant  points  in  the  cable  and  of  the  current  flowing  out  at 
the  receiving  end. 

We  may  regard  the  dot  signal  as  created  by  applying  to  the 
sending  end  a  source  of  positive  potential  and  keeping  it  on  for 
an  infinite  time,  but  after  the  lapse  of  a  time  T  superimposing 
upon  that  state  the  application  of  an  equal  source  of  negative 
potential  which  reduces  the  sending  end  to  zero  and  keeps  it 
zero. 

We  have  seen  that  the  effect  at  distant  points  in  the  cable  of 
applying  a  potential  +  V  at  the  sending  end  is  to  raise  the 
potential  at  a  point  at  a  distance  x  after  a  time  t  to  a  value 
r  =  V  $  (x,  t).  Hence  the  effect  of  applying  a  negative 
potential  —  V  after  the  lapse  of  the  time  T  is  represented  by 
r  =  —  V  $  (x,  (t  —  T)  ).  Hence  the  potential  in  the  cable  at 
any  distance  x  due  to  a  dot  signal  made  at  the  sending  end  is 
represented  by 

0»F{+(0,Q~+-(4'-'2)}      •  •    (32) 

Also  the  potential  due  to  a  dash  signal  is  represented  by 

v=V{<l>(x,(t-T))-<i>(x,t)}  .         .         .     (33) 


1GO        PROPAGATION   OF   ELECTRIC   CURRENTS 

Again,  we  have  seen  that  the  effect  of  applying  a  source  of 
potential  +  V  to  the  cable  at  the  sending  end  and  keeping  it  on 
is  to  cause  a  current  i  to  flow  out  at  the  receiving  end  which  is 

27 
represented  by  l=El  f(u>  ^' 

Hence  the  effect  of  making  a  dot  signal  at  the  sending  end 
must  be  to  cause  a  current  at  the  receiving  end  represented  by 

*'=^{  /(«,*)-/(«,(<-  2))}    •        -        •     (34) 

and  similarly  the  effect  of  making  a  dash  signal  at  the  sending 
end  must  be  to  cause  a  current  at  the  receiving  end  represented 

by 

/  («,*)}•  •     (35) 


We  can  therefore  select  any  combination  of  dot  and  dash 
signals,  in  other  words  any  letter  of  the  alphabet,  and  predict 
exactly  by  an  equation  the  current  which  will  at  any  instant  be 
found  at  the  receiving  end  of  the  cable  flowing  into  or  out  of  the 
earth.  The  expressions  (34)  and  (35)  are  in  fact  the  equations  to 
the  curves  representing  the  dot  and  dash  signals  as  recorded  at 
the  receiving  end  by  a  syphon  recorder  or  some  equivalent 
instrument. 

Thus,  for  instance,  let  us  consider  the  nature  of  the  received 
current  corresponding  to  a  dot  signal. 

We  may  consider  the  constant  factor  ZV/Rl  to  be  unity  and 

the  duration  T  of  the  dot  such  that  uT  =  ^-^  T  is,  for  example, 
0-3.  Then  we  have  0  =  *-*  and  Ol  =  f-«0-*>  =  t~ut  x  tuT  =  1-0, 
say.  Then  /  (u,  t)  =  \  -  0  +  04  -  6>9  +  6>16  -  <925,  etc.,  and 

/  (n,  (t  -  T))  =  \  -  Ol  +  Of  -  0!9  +  0!16  -  0!25,  etc. 

If  we  assign  to  ut  various  increasing  values,  0'4,  0'5,  0'6,  etc., 
we  can  calculate  the  values  of 

0  =  e-  ut  =  Cosh  ut  -  Sinh  ut, 


09  =  c-  9^  =  Cosh  9ut  -Sinh  9ut, 
and  so  on,  and  hence  obtain  the  value  of  f(u,  i)  in  the  form 


CURRENTS  IN  SUBMARINE  CABLES 


161 


/  (u,  t)  =  £  -Cosh  M£  +  Sinh  7^+Cosh  4?^  -Sinh  ±ut 

-  Cosh  9  ut  -{-  Sinh  9ut  +  Cosh  I6ut  -  Sinh  16^  -  etc.    .     (36) 
These  values  are  easily  obtained  from  any  good  table  of  hyper- 


of 


"T 


bolic   functions.       We   then   find    the   value 

equation  k  =  c"r  -—  Cosh  uT  -  Sinh  uT. 

Hence  0,  =  k  (Cosh  ut  -  Sinh  ut)  , 

Ol*  =  k*  (Cosh  4?^-  Sinh  4wQ,  etc. 
Therefore 

/  (w,  (t-T))  =    -k  Cosh  ut+k  Sinh  ^  +  A*  Cosh  hit-k*  Sinh 


from   the 


-A;9  Cosh  9ut+kQ  Sinh  9^,  etc.  .         .         .     (37) 

This  series  can  be  calculated  without  difficulty  by  means  of  a 
table  of  hyperbolic  functions  and  one  of  powers  of  e. 

It  is  then  easy  to  find,  by  subtracting  the  sums  of  the  two 
series  (36)  and  (37),  the  value  of  f(u,  t)  —f(u,  (t—T)  )  =f(ut,  T) 
for  various  values  of  ut. 

Thus,  if  uT  =  0*3,  the  following  values  of  the  above  function 
were  calculated  by  Everett  : 


ut. 

/OO-/(X-03). 

ut. 

/«*-/(«'  -0-3). 

04 

6 

2-3 

35 

0-5 

18 

2-4 

31 

0-6 

36 

2-5 

29 

0-7 

56 

2-6 

26 

0-8 

73 

2-7 

24 

0-9 

84 

2-8 

21 

1-0 

88 

2-9 

19 

1-1 

88 

3-0 

17 

1-2 

86 

3-1 

16 

1-3 

83 

3-2 

14 

1-4 

78 

3-3 

13 

1-5 

72 

3-4 

12 

1-6 

67 

3-5 

11 

1-7 

61 

3-6 

10 

1-8 

56 

3-7 

8 

1-9 

50 

3-8 

8 

2-0 

47 

3-9 

7 

2-1 

43 

4-0 

7 

2-2 

39 

E.C. 


162   PEOPAGATION  OF  ELECTEIC  CURRENTS 


The  curve  representing  the  above  values  or  the  "curve  of 
arrival  "  for  this  dot  signal  is  shown  plotted  in  Fig.  8.  It  will 
be  seen,  therefore,  that  the  effect  of  pressing  down  the  sending 


0-09 
0-03 


0-06 


W03- 

* 

0-02. 


0-01 


wt 

s " 

ElG.  8. — Curve  of  Arrival  of  Dot  Signal. 

key  for  a  short  time  and  applying  a  brief  constant  steady 
voltage  to  the  sending  end  appears  at  the  receiving  end  in  the 
form  of  a  current  which  rises  up  gradually  to  a  maximum  value 
and  then  fades  away.  Hence  these  dot  signals  cannot  be  repeated 


r 


0  T  2T  3T  4T  5T 

Time. 
"S"  Signed;. 

FIG.  9. — "  S  "  Signal  as  sent. 

faster  than  a  certain  limiting  speed,  or  else  the  effect  at  the 
receiving  end  is  indistinguishable  from  a  prolonged  dash  signal. 
We  here  see  the  reasons  for  the  limitation  of  the  speed  of  cable 
telegraphy.  The  larger  the  value  of  CRP  or  of  the  product  CR, 
viz.,  the  product  of  the  total  capacity  in  farads  and  resistance 


CURRENTS   IN   SUBMARINE   CABLES  163 

in  ohms  of  the  cable,  the  smaller  the  value  of  u,  and  the  longer 
will  be  the  time  before  the  current  at  the  receiving  end  reaches 
its  maximum  value  after  the  sending  key  is  depressed.  Also, 
the  smaller  the  value  of  u,  the  less  will  be  the  maximum  value 
of  the  received  current,  and  in  general  the  less  quickly  can  the 
intermittent  signals  succeed  each  other  consistently  with  retaining 
an  interpretable  form  at  the  receiving  end. 

The  above  method  of  calculation  enables  us  to  predict  the 
form  of  the  curve  representing  the  received  current  as  a  function 
of  the  time  for  any  assigned  signal  made  with  the  key  at  the 
sending  end.  Thus,  for  instance,  take  the  letter  S.  This  is 


V 


o-J       <>;>        (>•;>,        M       05       o  <>       0-1        O-S      0-9        1-0 

TVrrve  in    xccondLs. 

FIG.  10. — The  dotted  line  represents  the  "  S  "  Signal  as  sent,  and  the 
firm  lines  as  received  on  Cables  of  various  CR  values,  and  lengths. 
For  Curve  II.  length  ==  1,000  miles,  CR  =  1-0,  and  for  Curve  III., 
length  =  1,581  miles,  CR  =  2-5. 

represented  in  the  International  Morse  Alphabet  by  three  dots, 

each  space  between  the  dot  signals  being  equal  in  duration  to 

that  of  the  dot.    Hence  to  make  this  signal  the  key  at  the  sending 

end  is  tapped  three  times,  and  this  applies  to  the  sending  end  of  the 

cable  a  variation  of  potential  F,  represented  by  the  curve  in  Fig.  9. 

Let  the  duration  of  each  dot  and  each  space  be  represented  by 

T.     Then  the  current  at  the  receiving  end  is  expressed  as  a 

function  of  the  time  by  the  equation 

2F| 


(38) 


164        PROPAGATION   OF   ELECTKIC   CURRENTS 

To  calculate  Ir  we  have  to  give  to  the  symbol  t  various 
increasing  values,  0*1,  0'2,  0*3,  etc.,  and  calculate  the  value  of  the 
function  on  the  right-hand  side  of  the  expression  (38).  To  do 
this  we  must  have  the  length  of  the  cable  I,  the  sending  voltage 
F,  and  the  capacity  C  and  resistance  R  per  mile  given.  We  can 

then  calculate  -™    and  u  —  n--     Also  the  value  of  T  must 


be  given  in  fractions  of  a  second,  so  that  uT  is  known. 

With  some  considerable  labour  the  value  of  Ir  for  various 
values  of  t  can  be  calculated  and  the  curve  of  arrival  for  the 
S  signal  graphically  depicted.  This  has  been  done  for  the 
author  by  Mr.  G.  B.  Dyke  as  shown  in  Fig.  10,  which  represents 
the  form  of  the  curve  of  arrival  for  an  S  signal  on  certain 
hypothetical  cables. 

6,  The  Speed  of  Signalling  :  Comparison  of 
Different  Cables.  —  Every  type  of  receiving  instrument 
used  for  recording  telegraphic  signals  is  characterised  by 
requiring  a  certain  minimum  current  to  actuate  it.  Hence,  in 
order  that  the  particular  instrument  used  may  record  a  legible 
signal,  it  must  be  traversed  by  a  current  of  not  less  than  this 
critical  value  and  for  a  certain  period  of  time.  We  have  seen 
that  the  current  at  the  receiving  end  of  the  cable  is  a  function  of 
the  quantity  ut.  For  the  same  value  of  ut  and  for  the  same 
mode  of  working  or  making  the  signal  the  current  at  the 
receiving  end  will  be  the  same. 

It  is  therefore  necessary  to  have  a  particular  minimum  value 
of  ut  below  which  no  signal  will  be  recorded.  Accordingly  this 
value  of  ut  may  be  taken  as  a  working  constant.  Now  the  cable 

2 

has  a  particular  value  of  u  =  (jwn>  which  is  characteristic  of 

it,  and  hence  the  time  required  to  establish  the  minimum  or 
necessary  working  current  at  the  receiving  end  for  a  given  cable 
and  impressed  voltage  varies  inversely  as  u  or  directly  as  CUP. 
Hence  for  cables  made  in  the  same  manner,  but  of  various 
lengths,  this  time  varies  as  the  square  of  the  length.  The  speed 
of  signalling  varies  inversely  as  the  time  required  for  the 
received  current  to  reach  the  minimum  strength,  as  it  is  clear 


CUEEENTS  IN  SUBMAEINE  CABLES     165 

the  signals  cannot  succeed  each  other  more  frequently  than  N 
per  second  where  1/iY  is  the  time  required  to  affect  the  receiving 
instrument.  Hence  the  signalling  speed  varies  inversely  as  the 
product  CRl2  and  inversely  as  the  square  of  the  length  for  cables 
of  the  same  make. 

This  means  that  there  is  no  definite  "  velocity  of  electricity." 
The  interval  of  time  which  elapses  between  closing  the  circuit  at 
the  sending  end  and  recording  the  signal  depends  not  only  on  the 
sending  voltage,  but  upon  the  nature  of  the  receiving  instrument 
and  upon  the  length  of  the  cable.  This  explains  how  it  is  that 
the  older  electricians  and  telegraphists  obtained  such  very 
various  and  different  results  in  their  endeavours  to  measure  the 
supposed  velocity  of  electricity  along  a  wire  or  cable. 

The  speed  of  signalling  can  be  increased  by  decreasing  the 
total  resistance  and  total  capacity  of  the  cable.  This  latter, 
however,  is  not  much  under  control,  as  it  is  determined  chiefly 
by  the  dielectric  constant  of  the  insulator  which  is  used,  and  for 
submarine  cables  no  substance  has  yet  been  found  to  take  the 
place  of  gutta-percha.  Accordingly  the  increase  in  speed  chiefly 
depends  upon  an  increase  in  the  diameter  of  the  copper 
conductor.  Long  cables  must  therefore  necessarily  be  heavy 
cables  if  we  are  to  preserve  reasonable  speed  in  signalling.  An 
empirical  rule  for  speed  of  signalling  is  given  in  Mr.  Jacobs' 
article  "  Submarine  Telegraphy  "  in  the  Encyclopedia  Britannica 
(supplement  to  the  tenth  edition)  as  follows  :  If  S  is  the  number  of 
five-letter  words  which  can  be  sent  per  minute  through  a  cable 
when  using  the  Kelvin  syphon  recorder  as  receiver,  and  if  C  is 
the  total  capacity  and  R  the  total  resistance  of  the  cable,  then 

120 
S  =-         .     The  capacity  must  be  measured  in  farads  and  the 


resistance  in  ohms. 

For  example,  suppose  a  cable  8,142  nautical  miles  or  nauts 
in  length  to  have  a  resistance  of  three  ohms  per  naut  and  a 
capacity  of  0'33  microfarad  per  naut.  Then 


=~x  (3,142)2  =  9-87, 
and  u  =  =  1,  since  7r2  =  9*87  nearly. 


166        PROPAGATION   OF   ELECTRIC   CURRENTS 

120 
Hence  by  the  above  rule  S  =  -^TJ-=  12  —  13,  and  the  sending 


speed  would  be  twelve  to  thirteen  five-letter  words,  or  sixty  to 
sixty-five  letters  per  minute. 

We  are  therefore  able  to  predict  not  only  the  form  of  the 
current  curve  at  the  receiving  end  for  a  given  kind  of  signal 
made  at  the  sending  end,  but  also  the  speed  with  which  the 
signals  can  succeed  each  other  in  cables  with  various  values  of 
C,  R,  and  I. 

7.  Curb-sending. — It  will  be  clear  from  the  above 
explanations  that  the  obstacle  to  signalling  speed  is  the  effect 


0-03 


0 


FIG.  11. — Curve  of  Arrival  for  Curbed  Dot  Signal. 

of  the  capacity  and  resistance  of  the  cable  in  dragging  out  a 
sharply  made  signal  or  voltage  change  made  at  the  sending  end 
into  a  slow  rise  and  fall  of  current  at  the  receiving  end.  Hence 
until  the  cable  is  cleared  of  a  previous  signal  another  one 
cannot  be  usefully  despatched,  or  if  it  is  the  two  run  together 
into  a  received  signal  indistinguishable  as  two. 

One  method  by  which  speed  of  signalling  can  be  increased  is 
by  means  of  curb-sending. 

By  this  method  in  sending  a  dot  signal  the  cable  at  the 
sending  end  is  first  raised  a  positive  potential  for  a  certain  time, 
then  lowered  instantly  to  an  equal  negative  potential,  and  after 
about  two-thirds  of  the  above  time  put  again  to  earth.  In  other 
words,  we  send  into  the  cable  a  current  in  one  direction  and  then 


CURRENTS  IN   SUBMAEINE   CABLES 


167 


follow  it  instantly  by  another  in  the  opposite  direction  for  a 
somewhat  shorter  time.  The  effect  of  this  is  to  clear  the  cable 
more  quickly  for  the  following  signal. 

The  operation  at  the  sending  end  may  be  represented  by  a 
rectangular  line,  which  shows  the  application  of  a  positive 
potential  to  the  cable  followed  by  an  equal  negative  potential 
for  a  shorter  time,  and  then  by  an  earthing  or  reduction  to  zero 
potential. 

Let  us  consider  then  the  effect  of  the  above  operation  carried 
out  at  the  sending  end  upon  the  cable  at  other  different  points. 

If  +  V  and  —  V  are  the  positive  and  negative  potentials 
applied  to  the  sending  end,  the  former  for  a  time  T\  and  the 
latter  for  a  time  T2  --  rl\,  then  the  potential  v  at  any  distance  x 
along  the  cable  at  any  time  t  is  given  by 

v  =  V{4>(x,t)  -  ^(Xl(t  -  T,))  +  +(x(t  - 
and  the  received  current  by 


Thus,  for  instance,  if  the  value  of  url\  =  0*3  and  uTz  =  0'5, 
then  the  values  of  the  received  current  have  been  calculated  by 
Professor  Everett  on  the  assumption  that  the  factor  2  V/Rl  —  1 
for  various  values  of  ut  as  follows  : 


*t. 

/(„£)  _  2f(ut  -  0-3)  +/(«*  -  0-5). 

ut. 

/(«0  -  2/(«rf  -  0-3)  +f(ut  -  0-5). 

0-4 

6 

1-5 

15 

0-5 

18 

1-6 

13 

0-6 

35 

1-7 

11 

0-7 

50 

1-8 

10 

0-8 

56 

1-9 

9 

0-9 

53 

2-0 

8 

1-0 

44 

2-1 

8 

1-1 

34 

2-2 

7 

1-2 

27 

2-3 

5 

1-8 

24 

2-4 

5 

1-4 

20 

2-5 

5 

If  these  values  are  plotted  out  we  obtain  a  curve  of  the  form 
shown  in  Fig.  11. 


168        PKOPAGATION   OF  ELECTEIC   CUEEENTS 

On  comparing  it  with  the  curve  in  Fig.  8  representing  the 
uncurbed  signal  it  is  seen  that  the  uncurbed  signal  rises  more 
slowly  and  dies  away  more  slowly,  but  it  has  a  larger  maximum 
value  than  the  curbed  signal. 

It  is  found  that  if  condensers  are  inserted  in  series  with  the 
cable  both  at  the  sending  and  receiving  end  the  effect  is  to  curb 
the  signals  to  a  considerable  extent.  In  modern  practice  the 
cable,  however,  is  nearly  always  duplexed,  that  is  to  say  arranged 
with  an  artificial  line  of  equal  total  capacity  and  resistance  in 
the  manner  shown  in  Fig.  12. 

In  this  case  C\  and  C2  are  two  large  condensers.  C  is  the 
cable,  and  C3  is  an  artificial  line  which  consists  of  sheets  of 
tinfoil  placed  on  one  side  of  sheets  of  paraffined  paper,  the 


FIG.  12. — Arrangements  for  Duplex  Transmission  in  a  Submarine  Cable. 

opposite  side  of  the  paper  sheet  being  coated  with  a  strip  of 
tinfoil  cut  in  zigzag  fashion.  The  zigzag  tinfoil  strip  has 
resistance  and  capacity  with  respect  to  the  other  sheet  of  metal, 
which  is  earthed.  Such  a  line  can  be  adjusted  to  represent  a 
cable  of  any  length  and  of  any  capacity  and  resistance  per  unit 
of  length.  The  receiving  instrument,  generally  a  syphon 
recorder  r,  is  connected  between  the  ends  of  the  real  and 
artificial  cable,  and  another  condenser  <75  is  placed  in  series 
with  it.  The  battery  B  and  sending  key  K  are  joined  in  as 
shown.  The  artificial  line  can  so  be  balanced  against  the  real 
line  that  on  depressing  a  key  the  current  flows  equally  into  the 
two  condensers  C\  and  C%  and  into  the  real  and  artificial  lines, 
and  the  points  a  and  b  remain  at  the  same  potential.  Hence 
the  current  sent  out  through  the  cable  does  not  affect  the  local 
receiving  instrument. 

On  the  other  hand,  if  a  current  arrives  it  flows  to  earth  partly 


CURRENTS  IN  SUBMARINE  CABLES 


169 


£  -as 

;3    <D  O 

O'JS'o 

J->    O 


si:  bo 
2^.2 

nj    co 


111 


'oj    fn 


170       PKOPAGATION   OF   ELECTEIC   CURRENTS 

through  the  receiving  instrument  and  the  artificial  line  and  partly 
to  earth  through  the  local  battery.  The  cable  is  then  duplexed, 
and  signals  can  be  sent  and  received  at  the  same  moment. 

It  is  now  usual  to  dispense  with  the  condenser  C5  in  series 
with  the  recording  instrument  and  in  place  of  it  to  insert  an 
inductive  shunt  L  across  the  terminals  of  the  coil  of  the  syphon 
recorder.  The  effect  of  this  inductive  shunt  is  to  curb  the  signals 
and  clear  the  cable  quickly  for  the  next  signal.  The  sudden  quick 
rise  of  potential  at  the  terminals  of  the  recorder  which  accom- 
panies the  reception  of  the  first  part  of  the  signal  affects  the 
recorder,  but  the  slow  fall  which  takes  place  after  the  maximum 
is  past  causes  a  current  to  flow  through  the  inductive  shunt,  and 
the  recorder  coil  falls  back  quickly  to  zero. 

In  the  case  of  a  short  cable  or  one  with  small  CR  the  signals 
made  by  the  syphon  recorder  are  sharp  and  well  defined.  The 
syphon  recorder  consists  of  a  light  coil  of  insulated  wire  hung 
by  a  bifilar  suspension  in  the  field  of  a  strong  magnet  like  a 
movable  coil  galvanometer.  To  this  coil  is  attached  a  light  glass 
pen,  the  point  of  which  rests  on  a  strip  of  paper  tape  which  is 
moved  by  clockwork  beneath  the  pen.  If  then  the  coil  is  at  rest 
the  pen  traces  a  straight  line  along  the  centre  of  the  tape.  If  a 
brief  current  from  the  cable  is  sent  through  the  coil  the  latter  is 
jerked  on  one  side,  and  when  the  current  ceases  it  falls  back  to  its 
normal  position. 

The  effect  is  to  make  a  dot  signal  which  is  a  square  notch  on 
the  line  if  the  cable  is  very  short.  If,  however,  the  current  rises 
up  slowly  and  falls  again  slowly,  then  the  ink  line  is  a  rounded 
mark.  The  dash  is  made  by  reversing  the  direction  of  the 
current  and  therefore  of  the  motion  of  the  pen.  In  the  case  of 
short  cables  the  alphabetic  signals  made  by  groups  of  these  dots 
and  dashes  are  quite  legible,  but  in  the  case  of  long  cables  it 
requires  some  skill  to  guess  the  meaning,  since  the  marks  on  the 
tape  are,  as  it  were,  parts  of  "  curves  of  arrival  "  running  into  each 
other.  The  reproductions  of  syphon  recorder  tapes  in  Fig.  13 
are  from  experiments  kindly  made  for  the  author  by  Mr.  H. 
Tinsley  with  artificial  lines  of  different  capacities  and  resistances 
to  show  this  rounding  effect  on  the  signals  with  increasing  values 
of  CR. 


CHAPTER   VI 

THE   TRANSMISSION   OF   HIGH    FREQUENCY   AND    VERY    LOW 
FREQUENCY  CURRENTS  ALONG  WIRES 

1.  The  Modifications  in  the  General  Equation 
for  Transmission  in  the  Cases  of  very  High  and 
very  Low  Frequency.—  Returning  to  the  general  equation 
for  the  transmission  of  electrical  disturbances  along  a  cable,  we 
can  write  it  in  the  form 


•    •    •  a) 

where  v  is  the  potential  in  the  cable  at  a  point  at  a  distance  x 
from  the  sending  end  and  at  a  time  t. 

The  above  is  the  general  equation  for  the  propagation  of 
potential  changes  of  any  type  along  a  cable  having  resistance, 
capacity,  inductance,  and  leakage.  It  may  be  called  the  telephone 
equation.  It  has  been  fully  discussed  in  Chapter  IV.  Secondly, 
if  the  cable  is  such  that  L  and  S  are  very  small  relatively  to  R 
and  C  and  if  the  frequency  is  low  we  can  neglect  the  terms 
involving  L  and  S  and  write  the  equation  in  the  form 

d~v    -pr  dv  fo\ 

-5—  a  =  .n/C  -jT     .....      (2) 

dx2  dt 

This  is  the  case  of  the  submarine  telegraph  cable,  and  the 
above  equation  (2)  may  therefore  be  called  the  telegraph  equation. 
In  this  form  it  has  been  considered  in  Chapter  V.  Thirdly,  if 
R  and  S  are  very  small  or  negligible  and  if  the  frequency  is  very 
high  we  can  neglect  the  terms  involving  R  and  S  and  write  the 
equation  (1)  in  the  reduced  form 

d*v       T  d*v 

3&=CLW  .....    (3) 

Since  this  applies  in  the  case  of  electric  oscillations  or  very 
high  frequency  alternating  currents  as  employed  in  wireless 


172        PROPAGATION   OF   ELECTKIC   CUREENTS 

telegraphy,  we  may  call  the  above  equation  (3)  the  radiotelegraph™ 
equation. 

Lastly,  if  the  line  is  an  aerial  line  of  small  capacity  and  induct- 
ance operated  at  low  frequency  or  with  continuous  current  so 
that  the  principal  constants  are  the  resistance  R  and  leakage  S 
we  can  neglect  L  and  C,  and  the  general  equation  reduces  to 


Since  this  applies  in  the  case  of  lines  operated  at  very  low 
frequency  or  with  continuous  currents  and  with  such  high  voltage 
as  to  make  the  leakage  important,  we  may  call  the  above  equation 
the  leaky  line  equation. 

Furthermore,  if  the  variation  of  potential  with  time  is  simply 
harmonic,  that  is  if  the  applied  electromotive  force  is  a  simple 
sine  curve  E.M.F.,  then,  neglecting  the  effects  at  first  contact, 
we  can  say  that  after  a  short  time  the  variation  of  potential  is 
simply  harmonic  everywhere  and  varies  as  the  real  part  of  tjpt. 

Hence  jj=JPv  and  ^  =  -p*v.       Accordingly  the  equations  (1), 
(2),  (3),  and  (4)  above  then  take  the  form 

.        .     (5) 


fa\ 
d'2V 


......     (8) 

dx* 

We  have  already  discussed  the  equations  (1)  and  (;2)  and  (5) 
and  (6)  in  Chapters  IV.  and  V.,  dealing  with  telephony  and  sub- 
marine cable  telegraphy.  Hence  we  need  not  say  more  about 
them.  The  equations  (3)  and  (7)  and  (4)  and  (8)  remain, 
however,  to  be  discussed. 

2.  The  Propagation  of  High  Frequency 
Currents  along  Wires.—  Taking,  then,  the  equation  (3), 
viz., 


HIGH  FEEQUENCY  CURRENTS   ALONG  WIRES    173 

we   find   that   one   particular   solution   applicable   to   the   case 
considered  is 


For  if  we  differentiate  the  above  expression  (10)  twice  with 
regard  to  x  and  twice  with  regard  to  t,  we  find  that  when  the 
last  expression  is  multiplied  by  GL  it  is  the  same  as  the  former. 


d*o         A* 

and  d^=-cL 

Hence  (10)  is  a  solution  of  (9). 

We  see  that  it  implies  that  v  is  periodic  in  space,  that  is,  along 
the  wire  as  well  as  with  time.  Therefore,  in  the  case  of  a  wire 
traversed  by  a  high  frequency  current,  at  any  one  instant  the 
potential  varies  along  the  line  in  a  simple  harmonic  manner. 
If,  however,  we  fix  attention  upon  the  variation  of  potential  at 
any  one  point  in  the  line,  it  is  also  periodic  or  varies  as  a  simple 
cosine  function  of  the  time. 

If  we  substitute  #+-?  f°r  x  m  the  expression  (10),  whilst 

keeping  t  constant,  we  see  that  its  value  remains   unaltered, 
because  Cos*  (0  +  2?r)  =  Cos  0.     Hence  at  distances  along  the 

line    equal    to  =  A    the     potential     value    repeats    itself. 

Accordingly  this  distance  is  the  wave  length  of  the  potential 
along  the  line.     If  we  keep  x  constant  and  substitute  t~\  ---  -j— 

for  t  in   (10)   we  see  that  its  value  also   remains   unchanged. 
Hence  at  any  one  point  in  the  line  the  values  of  the  potential 

repeat  themselves  at  intervals  of  time  equal  to   T  =  -  —  j  —  • 

This  is  therefore  the  periodic  time  of  the  potential  variation. 
The  velocity  W  with  which  the  wave  of  potential  travels  is 

given    by    W  =  *.      Hence,   since   A  =  *?  and    T  =  **^L9 
we  have 


174        PKOPAGATION   OF   ELECTRIC   CURRENTS 

If  then  we  apply  at  the  end  of  a  very  long  wire  having  induct- 
ance L  and  capacity  C  per  unit  of  length  a  simple  periodic  high 
frequency  electromotive  force,  the  effect  will  be  to  make  waves 
of  electric  potential  travel  along  the  wire  with  a  velocity  1/VOL 
centimetres  per  second,  and  at  any  one  point  in  the  line  there 

will  be  oscillations  of  potential  with  a  frequency  • 

A. 

3.  Stationary   Oscillations   on    Finite   Wines. 

We  are  not  much  concerned  practically  with  the  propagation 
of  high  frequency  currents  along  extremely  long  lines,  but  when 
the  wires  are  of  length  less  than  or  comparable  with  the  wave 
length  we  may  have  the  phenomena  of  stationary  waves  pre- 
sented. Thus  suppose  a  thin  wire  of  not  very  great  length, 
having  a  capacity  C  and  inductance  L  per  unit  of  length,  to 
have  a  high  frequency  electromotive  force  applied  in  the  centre, 

the  frequency  n  being  such  that  the  quotient  of  W  =  -r=-  by  n, 

1 
or     jTffj  is  e(lual  to  about  twice  the  length  of  the  wire.    Then 

a  wave  of  potential  would  run  outwards  in  each  direction 
and  be  reflected  at  the  open  ends  of  the  wire  and  return  again  to 
find  that  the  electromotive  force  had  changed  its  phase  by  half 
a  period.  The  oscillations  of  electromotive  force  are  thus  in  step 
with  the  movements  of  the  wave  of  potential,  and  therefore  the 
latter  are  maintained  and  amplified.  The  whole  process  is 
exactly  like  that  by  which  stationary  oscillations  are  maintained 
on  a  rope  fixed  at  one  end  by  administering  little  jerks  to  the 
other  end  when  held  in  the  hand.  The  frequency  of  the  jerks 
must  agree  with  the  interval  of  time  taken  by  the  wave  motion 
to  run  along  the  rope  and  return. 

Moreover,  if  we  make  jerks  more  quickly,  say  twice  as  quickly, 
the  cord  can  accommodate  itself  to  this  increased  frequency  by 
dividing  itself  into  two  vibrating  sections  separated  by  a 
stationary  point  called  a  node,  each  loop  or  ventral  segment 
being  half  the  length  of  the  cord. 

In  the  same  manner  an  experienced  violinist,  by  lightly 
touching  a  string  at  one  point  and  bowing  at  another,  can  cause 
the  string  to  vibrate  in  sections  and  give  out  musical  notes  which 


HIGH  FREQUENCY   CURRENTS  ALONG  WIRES     175 

are  harmonics  of  the  fundamental  vibration.  An  exactly  similar 
phenomenon  can  be  exhibited  electrically. 

4.  The  Production  of  Loops  and  Nodes  of 
Potential  in  a  Conductor  by  High  Frequency 

Electromotive  Forces. — To  obtain  a  conductor  suitable 
for  exhibiting  these  effects  in  a  convenient  space  we  require  a 
conductor  along  which  waves  of  electric  potential  travel  rather 
slowly. 

In  the  case  of  ordinary  straight  single  wires  of  good  con- 
ductivity, waves  of  electric  potential  travel  along  the  wire  with 
the  speed  of  light,  or  about  1,000  million  feet  per  second.  If, 
therefore,  we  can  create  high  frequency  oscillations  having  a 
frequency  of  one  million,  the  length  of  the  wave  of  potential 
would  be  1,000  feet  or  so,  and  we  should  require  a  wire  500  feet 
long  to  exhibit  the  phenomena.  If,  however,  we  coil  a  fine  silk- 
covered  wire  on  an  ebonite  rod  so  as  to  form  a  long  helix  of  one 
layer  of  closely  adjacent  turns,  we  can  make  a  conductor  which 
will  have  a  capacity  of  approximately  the  same  value  per  unit  of 
length  as  a  metal  cylinder  of  the  same  dimensions  as  the  helix, 
but  an  inductance  per  unit  of  length  much  larger  than  that 
of  any  single  wire. 

If  a  long  helix  of  insulated  wire  is  made  as  above  described 
such  that  the  length  is  at  least  fifty  times  the  diameter,  the 
inductance  per  unit  length  of  the  helix  will  be  (irl)N)2  absolute 
electromagnetic  units  of  inductance,  that  is,  centimetres,  or 

JQ^-  (irDN)2  henry s,  where  D  is  the  mean  diameter  of  the  helix 

and  N  the  number  of  turns  of  wire  per  unit  of  length  of  the  helix. 

The  capacity  of  such  a  helix  will  depend  on  its  proximity  to 

the  ground,  but  if  placed  say  50  cms.  above  a  table  it  will  be  given 

1-5* 
approximately  by  the  expression     ^      21' 

It  will  be  found  on  trial  that  it  is  easy  to  construct  a  helix 
along  which  electric  waves  of  potential  will  travel  so  slowly  that 
for  frequencies  of  one  million  or  so  the  wave  length  will  bear 
comparison  with  such  lengths  of  helix  as  can  be  conveniently 
constructed. 


176   PROPAGATION  OF  ELECTEIC  CUEEENTS 

Thus,  for  instance,  on  a  round  ebonite  rod  about  2-J  metres 
long  the  author  wound  a  spiral  of  silk-covered  No.  30  S.W.G. 
copper  wire  in  a  helix  of  one  single  layer  215  cms.  long  and  having 
5,470  turns.  The  helix  had  a  mean  diameter  of  4'75  cms. 

The  inductance  L  of  such  a  helix  per  unit  of  length  is  then 
given  by 

T     /3-1415x4-75x5470\2 

\ 215  -j  =0-149 xlO6  cms. 

The  capacity  per  unit  of  length  calculated  by  the  formula 
3 

21  gave    C  =  0*187  X  10" 6   microfarads,  and  by   actual 

4    loge  5 

measurement  was  found  to  be  0*21  X  10 ~G  microfarads  when  the 
helix  was  supported  horizontally  and  50  cms.  above  a  table. 
The  velocity  of  propagation  of  a  wave  of  electric  potential  along 

this  helix  is  then  equal  to  1/VCL,  where  L  =  ^r-= — =^3  henry 

45 
and  C  =  OTK — Tni2  farad,  and  hence 


1        215  x  A/1000  xlO6 

W=— -7=  =  —    — ,  —  =  174  x  10b  cms.  per  second. 

VCL  V  45x32 

The  velocity  of  light  is  30,000  X  106  cms.  per  second,  and  hence 
the  velocity  of  a  wave  of  potential  along  the  above  helix  is  only 
1/172  part  of  that  of  the  velocity  of  light. 

If  then  we  apply  to  the  end  of  such  a  helix  a  high  frequency 
alternating  electromotive  force  having  a  frequency  of  about 
200,000  per  second,  the  result  will  be  to  create  a  wave  of  potential 
which  travels  a  distance  of  four  times  the  length  of  the  helix  in 
the  time  of  one  complete  oscillation.  For,  the  velocity  of  propa- 
gation being  174  X  106  cms.  per  second  and  the  frequency 
2  X  105,  the  corresponding  wave  length  A  must  be  870  cms.,  which 
is  not  far  from  four  times  215. 

An  alternating  E.M.F.  of  this  frequency  is  best  obtained  by 
means  of  the  oscillating  discharge  of  a  condenser.1 

i  For  a  full  discussion  of  this  mode  of  discharge  the  reader  is  referred  to  the 
following  books  by  the  Author  :  "  The  Principles  of  Electric  Wave  Telegraphy  and 
Telephony,"  2nd  Edition,  Chapter  I.  (Longmans  &  Co.)  ;  "An  Elementary  Manual 
of  Radiotelegraphy  and  Radiotelephony,"  Chapter  I.  (Longmans  &  Co.). 


HIGH  FREQUENCY  CURRENTS   ALONG  WIRES     177 

If  a  condenser  or  Leyden  jar  of  capacity  C\  is  joined  in  series 
with  an  inductance  LI  and  with  a  short  spark  gap,  and  if  the 
spark  balls  are  connected  to  an  induction  coil,  oscillatory  dis- 
charges of  the  condenser  will  take  place  through  the  inductance 

coil  having  a  frequency  given   by  the  formula  n  =  -     /7rT- 

A  7T  V  Cj-L/j 

where  C\  is  measured  in  farads  and  L\  in  henrys,  or  else  by  the 

.  5-033  xlO6       ,         n    .  ,  . 

formula  n  —  -  ,  where  C\  is  measured  in  microfarads 

VClxLl 

and  LI  in  centimetres. 

Thus  the  capacity  of  the  condenser  used  was  0'005835  mfd. 
and  the  inductance  of  the  coil  was  110,000  cms.  The  frequency 
of  the  oscillations  set  up  was  therefore  0'197  X  106,  or  nearly 
200,000. 

If  the  above-mentioned  helix  is  connected  to  one  end  of  the 
inductance  coil  and  the  other  end  of  the  coil  is  to  earth,  as  shown 
in  Fig.  1,  then  the  oscillations  set  up  in  the  inductance  coil  by 
the  discharge  of  the  condenser  or  Leyden  jars  create  electric 
impulses  on  the  end  of  the  helix  AB  equivalent  to  the  action  of 
an  electromotive  force  having  a  frequency  of   197,000.      The 
helix  has  thus  produced  upon  it  stationary  waves   of  electric 
potential,  and  owing  to  the  cumulative  action  the  amplitude  of 
the  potential  variation  at  different  parts  of  the  helix  increases 
from  a  minimum  at  the  end  by  which  it  makes  contact  with  the 
condenser  circuit  to  a  maximum  at  the  free  end.     At  this  last 
place  the  amplitude  of  potential  variation  may  be  so  great  that 
it  reaches  a  value  at  which  sparks  and  electric  brushes  fly  off  the 
end  of  the  helix.     In  any  case  the  gradual  increase  along  the 
helix  can  be  proved  by  holding  near  the  helix  a  vacuum  tube  of 
the  spectrum  type  (see  Fig.  1)  filled  with  the  rare  gas  neon  or 
in  default  one  with  carbon  dioxide.     The  tube  glows  when  held 
in  a  high  frequency  electric  field,  and  the  brilliancy  of  the  glow 
will  be  found  to  decrease  as  the  tube  is  moved  from  a  place  near 
the  open  end  of  the  helix  to  a  place  near  the  end  at  which  it  is 
attached   to   the   condenser   circuit.      We   may   represent   this 
variation  of  potential  along  the  helix  by  drawing  a  cylinder  or 
double  line  to  denote  the  helix  and  a  dotted  line  in  such  position 
that  the  distance  between  the  dotted  line  and  the  line  representing 

E.G.  N 


178       PEOPAGATION   OF  ELECTRIC  CURRENTS 

the  helix  denotes  the  amplitude  of  the  potential  variation  at  that 
point  in  the  helix. 

An  analogy  is  found  in  the  case  of  a  strip  of  steel  held  at  one 
end  in  a  vice  and  made  to  vibrate  by  pulling  it  on  one  side  and 
letting  it  go.  The  amplitude  of  the  motion  of  the  different  parts 
of  the  strip  increases  from  zero  at  the  bottom  end,  where  it  is 
gripped,  up  to  a  maximum  at  the  free  end.  We  can,  however, 
make  the  above  steel  strip  vibrate  in  such  a  manner  that  there  is 
a  node  of  vibration  at  a  point  about  one-third  of  the  way  from 
the  free  end.  In  the  same  manner  if  we  decrease  the  capacity 


FIG.  1. — Arrangement  of  Apparatus  for  producing  stationary  electric 
oscillations  on  a  helix  A  B.  C,  C,  are  Leyden  Jars,  L  is  an 
inductance  coil,  and  S  is  a  spark  gap. 

and  inductance  in  the  condenser  circuit  to  which  the  helix  is 
attached  so  as  to  make  the  frequency  of  the  electromotive  force 
acting  on  the  end  of  the  helix  three  times  that  required  to  pro- 
duce the  fundamental  vibration,  or  say  about  600,000  in  the  case 
of  the  helix  above  described,  then  the  effect  will  be  that  to 
accommodate  itself  to  the  tripled  frequency  the  stationary  waves 
of  potential  on  the  helix  must  have  a  node  of  potential  at  about 
one-third  of  the  way  from  the  free  end,  and  the  distribution  of 
potential  amplitude  can  be  denoted  by  the  ordinates  of  the  dotted 
line  in  Fig.  2. 
In  the  same  manner  by  increasing  the  frequency  to  5,  7,  9, 


HIGH   FKEQUENCY   CUREENTS   ALONG  WIRES     179 

etc.,  times  that  required  to  excite  the  fundamental  oscillations  on 
the  helix,  we  can  create  harmonic  oscillations  whicli  have  2,  3,  4, 


°l  IE 

<: 2OO -a- 


uN DAM  ENTAL 


N, 


----  .50  -------  3><  ------------------  140  -----------------  >• 

IST  HARMONIC 


N,  ,—  ------  —  ,    N 


-  -86 


------  57  ------  ^<  ----  58  ------  ^<  ----  62  ------- 

ARMONIC 


3RD  H 


PJ         ^y-x ^'^ N^ ^-^ ii.r- 

°l  IL 

<-\Q-  $?*< 4  4 ^>< 44 X 46 ><• 48 >• 

A.~™   H  ARMON  1C 


^ 
° 


<-!5-><---36  ----  ><  ----  36---X---37---X---3  7  ----  ><-  ----  39  ---^> 

5™  HARMONIC 

LENGTHS    IN    CMS 

FIG.  2.—  Diagram  illustrating  the  formation  of  nodes  and  loops  of  potential 
upon  a  helix  by  means  of  electromotive  forces  of  progressively  increasing 
frequency. 

etc.,  nodes  of  potential.     The  existence  of  these  nodes  can  be 
proved  by  holding  a  neon  vacuum  tube  near  the  helix  and  moving 

N  2 


180   PBOPAGATION  OF  ELECTKIC  CURRENTS 

it  along  from  one  end  to  the  other.  When  near  a  node  the  tube 
will  not  glow,  but  when  opposite  to  an  antinode  or  ventral  segment 
it  will  glow  very  brightly. 

The  distance  between  two  adjacent  nodes  is  half  a  wave  length 
of  the  stationary  oscillations.  Hence  from  this  measured  wave 
length  A  and  the  calculated  speed  of  propagation  W  we  can 
determine  the  frequency  n  =  IF/A  and  prove  that  this  agrees 
with  the  frequency  of  the  condenser  circuit  which  excites  that 
oscillation.  In  the  case  of  the  helix  above  mentioned  the 
measurement  of  this  internodal  distance  for  two  consecutive 
nodes  for  the  various  harmonics  was  as  follows :  for  the  1st 
harmonic  140  cms.,  for  the  2nd  harmonic  86  cms.,  for  the  3rd 
harmonic  62  cms.,  for  the  4th  harmonic  48  cms.,  and  for  the 
5th  harmonic  39  cms.  These  distances  are  the  half  wave  lengths. 
Hence,  doubling  them,  we  have  280, 172,  124,  96,  and  78  for  the 
harmonic  series  of  observed  wave  lengths  A.  Correspondingly 
it  was  necessary  to  adjust  the  condenser  capacity  C\  and  induc- 
tance LI  so  that  the  frequencies  n  calculated  from  the  formula 

n  =  - — /-frr  gave  values  respectively  of 

ZTT  *  C/I.L/I 

0*588  X  106  to  produce  the  1st  harmonic, 

0*977  X  106  to  produce  the  2nd  harmonic, 

1*379  X  106  to  produce  the  3rd  harmonic, 

1*70     X  106  to  produce  the  4th  harmonic, 

1*9  X  106  to  produce  the  5th  harmonic. 
Taking  the  observed  values  of  the  wave  length  A  and  the 
calculated  values  of  the  frequency  n,  we  can  deduce  the  wave 
velocities  W=  ?iA,  and  these  are  respectively  165  X  106, 168  X  106, 
171  X  106,  163  X  106,  and  148  X  106.  The  mean  value  is 
163  X  106  =  W.  This  compares  fairly  well  with  the  calculated 
value  172  X  10G  determined  from  the  measured  capacity  and 
inductance  of  the  helix  per  unit  of  length,  having  regard  to  the 
small  value  of  these  last  quantities  and  consequent  difficulty  in 
measuring  them  exactly. 

It  is  sufficient  to  show  that  all  the  harmonic  oscillations 
travel  with  equal  velocity,  and  that  this  velocity  is  equal  to  the 
value  of  1/VCL,  where  C  and  L  are  the  capacity  and  inductance 
per  unit  of  length  of  the  helix. 


HIGH  FREQUENCY  CURRENTS  ALONG  WIRES    181 

The  condition  then  for  obtaining  stationary  electric  waves  on 
the  helix  is  that  the  time  taken  for  the  wave  to  run  twice  to 
and  fro  on  the  helix  must  bear  some  integer  ratio  to  the  period 
of  the  applied  electromotive  force.  If  I  is  the  length  of  the 
helix  and  W  the  wave  velocity,  then  the  time  taken  for  the  wave 
to  run  twice  there  and  back  along  it  is  41/W.  But  W  —  1/VCL. 
Hence  t  =  4lVCL. 

Suppose  then  that  the  time  period  of  the  applied  electro- 
motive force  is  T  =  4lVCL,  the  wave  will  travel  twice  to  and  fro 
in  this  time,  and  we  shall  have  the  ratio  T/£=l,  or  the  oscillation 
excited  will  be  the  fundamental  oscillation. 

The  wave  length  A  will  then  be  such  that  A  r=  WT  =  41,  or 
the  fundamental  wave  length  will  be  four  times  the  length  of 
the  helix,  or  4  X  215  =  SCO  cms. 

If,  however,  the  frequency  of  the  applied  electromotive   force 


_ 
is  three  times  greater,  or  TI  =  -jrCL,  then  the  ratio  T\ft  =  o, 

4:1 

and  the  wave  length  Ax  =  WT\  =  -^  .     If  the  frequency  of   the 

applied   electromotive    force    is  increased   respectively  to  5,  7, 
9,    11,    etc.,    times   that   required    to   create   the   fundamental 

oscillation,  we  shall  have  time  periods  7'2  =  jrVCL,   T3  =  y-  VCL, 

4:1  11 

Ti  =  9  VCL,   etc.,   and    ratios    rl\\t  —  -g,    T3/t  =  ^  ,   etc.,  and 

41  41  4:1  4:1 

therefore  wave  lengths  A2  =  v-,    A3  =  -=-,    A4  =  g-,    A^.=-JJ. 

In  the  case  of  the  helix  described  these  harmonic  wave 
lengths  should  therefore  be  860/3,  860/5,  860/7,  860/9,  860/11 
cms.,  or  286,  172,  123,  95,  and  79  cms.  respectively. 

But  the  observed  values  as  obtained  from  twice  the  internodal 
distances  were  280,  172,  124,  96,  and  78  cms.  respectively,  so 
the  observed  values  of  A2,  A3,  etc.,  agree  very  well  with  those 
which  theory  requires. 

Hence  any  such  helix  of  length  I  can  have  stationary  waves 
produced  upon  it,  fundamental  or  harmonic  oscillations  of  wave 

41  41  41  4:1  4:1 

length  A0  =  41,  Ax  =  —  ,    A2  =  -^,    A3  =  j,    X4  =  -g,    A5  =  .Q,  etc., 


182        PEOPAGATION   OF   ELECTEIC   CUEEENTS 

by  applying  to  its  end  alternating  electromotive  forces  of 
increasing  frequency  in  the  ratios  1,  3,  5,  7,  9,  etc. 

These  facts  have  application  in  wireless  telegraphy.  An 
essential  feature  of  the  arrangements  for  producing  the  electric 
waves  which  are  radiated  through  space  to  conduct  wireless 
telegraphy  is  a  long  wire  insulated  at  one  end  and  connected 
to  the  earth  or  to  a  balancing  capacity  at  the  other  end.  The 
wire  is  called  the  aerial  or  antenna.  At  some  point  near  the 
earthed  end  a  high  frequency  electromotive  force  is  applied  in 
the  wire,1  and  the  frequency  of  this  electromotive  force  is 
adjusted  with  reference  to  the  length  of  the  wire  so  as  to  produce 
stationary  oscillations  in  the  wire  subject  to  the  condition  that 
the  earthed  or  lower  end  must  be  a  node  of  potential  and  the 
upper  or  insulated  end  of  the  wire  a  loop  or  antinode  of  potential. 
We  can  therefore  set  up  oscillations  which  are  the  fundamental 
or  higher  harmonics,  and  which  have  frequencies  in  the  ratio  of 
1,  3,  5,  7,  9,  etc.  These  oscillations  on  the  wire  create  electric 
waves  in  the  space  around.  In  the  same  manner  we  can  set  up 
on  spiral  wires  stationary  oscillations  of  various  kinds.  The 
possible  types  of  oscillation  on  an  aerial  wire  or  antenna  as  used 
in  radiotelegraphy  are  illustrated  in  Fig.  2,  where  the  ordinates 
of  the  dotted  line  or  its  distance  from  the  thick  black  line, 
representing  the  antenna,  denotes  the  amplitude  of  the  potential 
oscillation  at  that  point  in  the  wire.2 

5,  The  Propagation  of  Currents  along   Leaky 

Lines. — Turning  then  to  the  fourth  reduced  case  of  the  general 
equation,  we  have  to  discuss  equation  (4)  for  the  case  in  which 
the  frequency  is  very  low,  or  the  current  even  continuous,  and 
the  inductance  and  capacity  small,  but  the  resistance  and 
leakance  large.  In  this  case,  when  the  quantity  pL  can  be 

1  For  details  sec  the  Author's  works  on  Wireless  Telegraphy,  "An  Elementary 
Manual  of  Radiotelegraphy  and  Radiotelephony, "  or  "The  Principles  of  Electric 
Wave  Telegraphy  and  Telephony  "  (Longmans,  Green  &  Co.,  39,  Paternoster  Row, 
London). 

For  further  information  on  the  production  of  stationary  fundamental  and 
harmonic  oscillations  in  wireless  telegraph  antennas  the  reader  is  referred  to  the 
Author's  book  "  The  Principles  of  Electric  Wave  Telegraphy  and  Telephony," 
Chapter  IV.,  2nd  Edition. 


HIGH  FREQUENCY  CURRENTS  ALONG  WIRES     183 

neglected  in  comparison  with  R  and  also  pC  in  comparison  with 
S,  the  general  equation  reduces  to 


Let  us  write  a2  for  RS.     Then  the  equation  becomes 

dto 

—~=a2v. 
dx2 

This  is  a  well-known  differential  equation,  which  is  satisfied  by 
v  =.  Aeax  or  v  =  B€~ax,  where  A  and  B  are  constants.  Hence 
the  solution  in  the  above  case  is 


Instead  of  eax  and  t~ax  substitute  in  the  above  equation  the 
equivalent  expressions, 

tax=  Cosh  a#-f  Sinn  ax'  and 

e-a*_  Cogh  ax  —  Sinh  ax. 
We  have  then  on  collecting  terms 

v  =  (A  +  B)  Cosh  ax  +  (A-  B)  Sinh  ax       .        .     (12) 

If  we  take  the  origin  at  the  sending  end  of  the  cable  and 

assume    that    an    electromotive    force    V\   is    applied    at    that 

end,    then   when  x  =   0  we  have  v  =   Fi,  but  when  x  —  0 

Cosh  ax  =  1,  Sinh  ax  =  0.     Hence  V\  —  A  +  5. 

Again,  the   current   i  at  any  point  in  the  line  is  equal  to 

~~'  smce  ^e  current  ig  measured  by  the  drop  in  potential 


down  a  length  dx  divided  by  the  resistance  of  that  length.     If 
we  differentiate 
for  the  current 


we  differentiate  (12)  and  multiply  by  —  -    we  have  the  expression 


Smliax-~(A-B)  Cosh  ax          .     (13) 

But  when  x  —  0  i  =  Ii  =  current  at  the  sending  end.    Therefore 
we  have 


and  also  A  -f-  B  —  V\. 

Substituting  these  values  of  A  +  B  and  A   -  B  in  (12),  we 
have 

v  =  F!  Cosh  ax-1—  Sinh  ax      .         .         .     (14) 


184        PEOPAGATION   OF   ELECTEIC   CUKRENTS 

,     .        .  1  dv         ~    -, 

and,  since  ^  =  —  ^  ^,  we  find 

V a 
i  =  I1  Cosh  ax — jj-  Sinh  ax  (15) 

Let  us  denote  the  insulation  resistance  of  the  line  per  mile  by 
r;  then  r  =  1/S,  and,  since  a  =  VRS,  we  have  a  =  \  —,  and 

substituting  this  value  of  a  in  (14)  and  (15),  we  arrive  finally  at 
the  expressions 

v =  F!  Cosh  ax -I^Wr  Sinh  ax  .         .     (16) 

i=Ij,  Cosh  ax—  -7^=-  Sinh  ax     .         .         .     (17) 

which  give  us  the  potential  v  and  current  i  at  any  distance  x 
from  the  sending  end  of  a  line  of  conductor  resistance  11  and 
insulation  resistance  r  per  unit  of  length. 

We  will  then  consider  various  cases  in  which  the  line  is 
(i.)  insulated,  (ii.)  earthed  at  the  far  end,  and  (iii.)  earthed 
through  a  receiving  instrument  of  known  resistance. 

(i.)  Line  insulated  at  the  far  end. — In  this  case  we  have  zero 
current  at  the  extremity.  Hence  in  equation  (17)  put  i  =  0  and 
x  =  I,  where  I  is  the  length  of  the  line ;  then 

/!  Cosh  aZ  =  ^=  Sinh  al         .         .         .     (18) 

or  /i  VRr=  FI  Tanh  al     .         .         .         .     (19) 

Substituting  from  equation  (19)  in  (16),  we  have 

v=  FjICosh  ax- Sinh  ax  Tanh  al}          .         .     (20) 
This  gives  us  the  potential  v  at  any  point  in  a  leaky  line. 
If  we  take  x  —  I,  then  (20)  becomes 

v=VlSechal          ....     (21) 
and  as  I  increases  v  continually  diminishes. 

If  the  line  had  no  leakage,  that  is  if  r  =  x  ,  then  we  should 
have  had  v  =  Fi  at  the  far  end  when  that  end  is  insulated. 
Also  from  (19)  and  (17)  we  find 

i  =  I1{Cosh  ax  —  Sinh  ax  Coth  al}  .         .         .     (22) 
which  gives  us  the  current  at  any  point  in  the  leaky  line. 

We  can  put  the  formulae  (20)  and  (22)  for  the  voltage  and 
current  in  a  simpler  form  if  we  measure  the  distances  from  the 


HIGH  FREQUENCY  CURRENTS  ALONG  WIRES    185 

free  end.    Let  xf  be  the  distance  of  a  point  from  the  free  end,  and 
let  x'  --—  I  —  x. 

Then  formula  (20)  is  equivalent  to 

i7  =  7rJj— = .Cosh  00'     .        .  .     (23) 

Cosh  al 

and  (22)  can  be  written 

x'     ....     (24) 


Sinh  al 

Hence  the  potential  at  any  point  in  the  leaky  line  is  pro- 
portional to  the  hyperbolic  cosine  of  ax'  and  the  current  to  the 
hyperbolic  sine  of  ax'.  Hence  when  x'  =  0  we  have 

v=  Fj/Cosh  al=  F!  Sech  al, 
as  before.     Let  us  consider  next, 

(ii.)  The  line  earthed  at  the  far  end. — Then  for  x  =  I  we  have 
v  =  0,  and  therefore  substituting  these  values  in  (16),  we  have 

I^~Br  Sinh  al=  Vl  Cosh  al  (25) 

and  substituting  this  last,  (25),  in  both  (16)  and  (17),  we  arrive  at 
the  equations 

v  =  FijCosh  ax  -  Sinh  ax  Coth  al}          .         ,     (26) 

i  =  /!{  Cosh  ax  —  Sinh  ax  Tanh  al}         .         .     (27) 

If  we  reckon  distances  from  the  earthed  end  and  let  x'  be  such 
distance,  so  that  x'  =  I  -  x,  then,  substituting  in  the  above 
formulae,  we  have 

v  =  Q.VS    7Sinbaa;'     ....     (28) 
Smh  al 

*•=_ A—  Cosh  ax'    ....     (29) 
Cosh  al 

Hence  at  the  earthed  or  receiving  end  the  current  is  given  by 

and  when  I  is  very  large  this  received  current  is  zero. 

We  have  then  to  consider  the  case 

(iii.)  When  the  line  is  earthed  through  a  receiving  instrument  of 
known  resistance. — We  shall  consider  that  the  receiving  instru- 
ment has  a  resistance  p  and  a  negligible  inductance.  Then  the 
current  through  the  receiving  instrument  is  72  =  T2/p. 


186        PKOPAGATION   OF   ELECTKIC   CUEKENTS 

Kef  erring  to  the  general  equations  (16)  and  (17), 

v=Vi  Cosh  ax  —  Jx  Vltr  Sinh  ax, 

y 
i=I1  Cosh  ax  —  —  r;=  Sinh  ax, 

we  put  x  =  Z,  and  we  have 

Fa  =  Ia  p=Fx  Cosh  aZ-Ij  v'Br  Sinh  aZ      .         .     (31) 


Ij  Cosh  aZ-  -/sinh  al  .         .         .     (32) 


Eliminating  Ii  from  these  two  last  equations  we  obtain 

y 
j  __  _  Y  i 

p  Cosh  aZ+  V  .Rr  Sinh  al 
Also  eliminating  I2,  we  have 

/Ifr  Cosh  aZ+p  Sinh  aZ 


(33) 

Cosh  aZ+  Vlfr  Sinh  al 
Consider  a  hyperbolic  angle  y  such  that  Tanh  y  =  p/Vlir,  and 

therefore       Sinh  y  —  -/==  and  Cosh  y  =  —         =. 
vBr  —  p1  vEr  —  p2 

Then  we  can  write  the  expressions  (33)  and  (34)  in  the  form 

/2=          Cosech  (a7+y)  '  (85) 


'         '         '     (36) 


On  comparing  the  above  expressions  with  those  given  in 
Chapter  III.  for  the  propagation  of  telephone  currents  in  a  line 
with  constants  E,  L,  C,  and  S,  it  will  be  seen  that  the 
expressions  are  similar,  but  that  the  quantity  Vlir  here  takes 
the  place  of  the  initial  sending  end  impedance  and  p  that  of  the 
impedance  of  the  receiving  instrument. 

The  ratio  of  the  received  to  the  sending  end  current  is 

'      '      '    '37) 

which  reduces  to  (30)  when  p  =  0.  All  these  expressions  are 
applicable  to  continuous  currents  flowing  in  leaky  lines.  For  a 
given  line  of  given  leak  per  mile  the  effect  of  placing  a  receiving 
instrument  at  the  receiving  end  is  equivalent  to  increasing  the 
length  of  the  line  by  an  amount  Z'  such  that 


CHAPTER  VII 

ELECTRICAL    MEASUREMENTS    AND    DETERMINATION    OF    THE 
CONSTANTS    OF    CABLES 

1.  Necessity  for  the  Accumulation  of  Data  by 
Practical  Measurements.— As  a  long  submarine  cable 
or  telephone  line  is  a  costly  article,  the  predetermination  of  its 
performance  is  a  matter  of  the  utmost  importance.  It  is 
therefore  necessary  to  bring  to  bear  upon  its  construction  and 
testing  a  large  knowledge  of  the  results  of  previous  constructions 
of  the  same  or  similar  cables.  This  requires  electrical  testing. 
In  fact,  we  may  say  that  out  of  the  attempts  to  lay  the  first 
very  long  submarine  cables  the  whole  of  our  practical  and 
absolute  system  of  electrical  measurements  has  arisen.  We 
have  to  determine  for  every  cable  and  line  the  primary  constants, 
viz.,  conductor  resistance,  inductance,  capacity,  and  the  insula- 
tion resistance,  all  per  statute  or  nautical  mile  or  kilometre,  and 
especially  measurements  of  the  attenuation  constants,  to  provide 
a  store  of  knowledge  on  which  we  can  draw  in  designing  other 
cables.  Experimental  means  are  therefore  required  for  accurately 
measuring  these  quantities  as  well  as  others,  such  as  line  and 
instrumental  impedances,  and  the  currents  and  phase  angles  to 
enable  forecasts  to  be  made  of  the  operation  of  proposed  lines 
or  cables  when  constructed  in  a  predetermined  manner.  For 
much  of  the  information  on  the  methods  of  electrical  measure- 
ments generally  the  reader  must  be  referred  to  existing  text- 
books, but  it  will  be  convenient  to  epitomise  some  of  the  most 
necessary  information  in  this  chapter.1 

1  The  reader  may  be  referred  to  a  treatise  by  the  Author  entitled  "A  Handbook 
for  the  Electrical  Laboratory  and  Testing  Room,"  2  vols.,  The  Mcctrir'uui  Printing 
and  Publishing  Company,  Ld.,  1,  Salisbury  Court,  Fleet  Street,  and  also  to  the 
well-known  work  by  Mr.  H.  R.  Kempe  on  '•  Electrical  Testing." 


188       PKOPAGATION  OF   ELECTEIC   CUEEENTS 

2.  The    Predetermination    of   Capacity.— Since  a 

telegraph  or  telephone  wire  is  only  a  long  cylinder  of  metal  or 
else  a  similar  structure  composed  of  stranded  wires  of  which  the 
section  is  approximately  circular,  we  have  first  to  consider  the 
capacity  of  such  a  long  cylinder  in  various  positions  with  regard 
to  the  earth  or  other  conductors. 

Definition. — The  electrical  capacity  of  a  body  is  measured  by 
the  quantity  of  electricity  or  charge  which  must  be  imparted  to 
it  to  raise  its  potential  by  one  unit  when  all  other  neighbouring 
conductors  are  maintained  at  zero  potential. 

Definition. — The  potential  at  any  point  due  to  any  charge  on 
an  extremely  small  conductor  at  any  other  point  is  measured 
by  the  quotient  of  the  small  charge  or  quantity  of  electricity  by 
the  distance  between  the  conductor  and  the.  point  in  question. 
Hence  if  we  have  any  small  charge  dq  on  a  conductor  the 
potential  at  a  distance  r  from  that  charge  is  dq/r.  The 
potential  due  to  a  finite  charge  is  the  sum  of  all  the  potentials 
due  to  the  elements  of  the  charge  respectively.  Thus  if  a  body 
has  a  charge  Q,  and  we  divide  it  into  elements  of  charge  dQ, 
then  the  potential  at  any  point  is  the  sum  of  all  the  quantities 
dQ/r,  where  r  is  the  distance  from  the  point  in  question  to  each 
element  of  the  total  charge. 

Two  other  facts  connected  with  electric  potential  and  charge 
are  (i.)  that  electric  charge  resides  only  on  the  surface  of 
conductors,  and  (ii.)  that  the  potential  of  all  parts  of  a  conductor 
is  the  same.  These  principles  enable  us  to  calculate  the 
capacity  of  conductors  of  a  certain  symmetry  of  form  in  simple 
cases.  For  example,  we  may  find  the  capacity  of  a  conducting 
sphere  as  follows :  Let  a  charge  Q  be  supposed  to  be  uniformly 
distributed  over  it,  and  let  it  be  assumed  to  be  divided  into 
elements  of  charge  dQ.  Let  the  radius  of  the  sphere  be  R. 
Then  the  potential  at  the  centre  of  the  sphere  due  to  each 
element  of  charge  is  dQjll,  and,  since  all  elements  are  situated 
similarly  with  regard  to  the  centre  of  the  sphere,  the  potential 
at  the  centre  of  the  whole  charge  is  Q/R.  But  this  must 
therefore  be  the  potential  V  of  any  point  in  the  sphere. 
Hence  Q/R  =  V  or  Q/V  =  R.  Now  the  ratio  of  charge  to 
potential  is  defined  to  be  the  capacity  C  of  the  conductor.  Hence 


THE  CONSTANTS  OF  CABLES        189 

for  such  a  sphere  C  =  It,  or  the  capacity  in  electrostatic  units  is 
numerically  equal  to  the  radius  of  the  sphere. 

Since  9  X  105  electrostatic  units  capacity  are  equal  to 
1  microfarad,  we  find  that  the  capacity  of  the  sphere  of 
radius  R  is  equal  to  11  / '(9  X  105)  microfarads,  where  R  is  measured 
in  centimetres. 

This,  however,  is  on  the  assumption  that  the  sphere  has  a 
uniformly  distributed  charge,  and  that  all  other  conductors 
are  at  a  very  great  distance.  The  actual  capacity  of  a  con- 
ducting sphere  of  radius  R  cms.  hung  up  in  a  room,  for 
instance,  would  be  found  to  be  somewhat  more  than  R/  (9  X  105) 
microfarads. 

For  instance,  let  a  conducting  sphere  be  surrounded  by  a 
concentric  spherical  shell,  and  let  the  radius  of  the  outer  surface 
of  the  inner  sphere  be  RI  and  that  of  the  inner -surf  ace  of  the 
outer  shell  be  11%.  Then  if  a  positive  charge  Q  is  placed  on  the 
inner  sphere  it  will  induce  an  equal  negative  charge  on  the 
inner  surface  of  the  outer  shell,  and  if  this  outer  shell  is  earthed 

the  potential  at  any  point  in  the  inner  sphere  will  be  ~----^-=  V, 

J*i      -"a 

O  7?  7? 

and  hence  ~~  =  C  =  •„  l_^  electrostatic  units,  or  the  capacity 

7?  7?  1 

of  the  inner  sphere  in  microfarads  will  be  p  l_^>   « — j^mfds., 

which  becomes  equal  to  ll\j  (9  X  105)  when  R%  is  infinite.  The 
capacity  of  the  sphere  is  therefore  increased  by  the  proximity 
of  another  conductor  even  though  the  latter  is  connected  to 
earth. 

In  the  same  manner  we  can  obtain  an  expression  for  the 
capacity  of  a  long  cylindrical  wire  of  circular  section.  Take  a 
point  0  on  the  central  axis  for  origin,  and  consider  any  element 
of  the  surface  cut  off  by  two  transverse  planes.  Let  the  radius 
of  the  circular  section  be  r,  and  the  axial  length  of  the  element 
be  bx,  and  the  axial  distance  of  the  elements  from  the  origin 
be  x.  Then  the  surface  of  that  element  is  %xrSxt  and  if  p  is 
the  surface  density  of  a  charge  uniformly  distributed  over 
the  wire,  the  charge  on  that  element  of  surface  is  ZirrpSx. 
The  distance  of  all  parts  of  this  element  of  charge  from  the 


190       PROPAGATION   OF   ELECTRIC   CURRENTS 

origin  is  Vr2  +  x2,  and  hence  the  potential  of  the  element  at 
the  origin  is 

Hence  the  potential  V  of  the  whole  charge  spread  uniformly 
over  a  wire  of  length  I  is  obtained  from  the  integral 

•    •  (2) 

r      J™  f  \ 

The  integral 

Hence  7=4wrP  |  loge  {  ^+^r*  +  ~)  -log.  r\  '     (3) 

But,  since  Q  —  %irrpl  is  the  whole  charge  on  the  wire,  the 
capacity  C  =  Q/V.  Therefore  we  have  for  the  capacity  of  the 
circular-sectioned  wire  of  length  I  and  diameter  cl  =  2r  the 
expression 

w 


—  log  r 


and  if  r  is  small  compared  with  -    this  becomes 


2  log, 


(5) 


The  above  formula  gives  the  capacity  in  electrostatic  units. 
If  we  use  ordinary  logarithms  and  reckon  in  microfarads  it 
becomes 

0(inmfds.)  =  -  —  07       •         •     (6) 

4-6052  x9x  10s  xlogM-^ 

The  length  I  must  be  expressed  in  centimetres. 

This  formula  is  useful  in  calculating  the  capacity  of  a  single 
vertical  wire  used  as  an  antenna  in  radiotelegraphy,  but  in 
practice  it  will  generally  give  a  value  about  10  per  cent,  or  so, 
too  small  on  account  of  the  proximity  of  the  antenna  wire  to  the 
earth.  The  formula  (4)  is  in  fact  the  capacity  of  a  wire  at  an 
infinite  distance  from  all  other  conductors. 


THE   CONSTANTS   OF   CABLES 


191 


Another  useful  expression  for  the  potential  of  a  long,  straight, 
thin-charged  wire  at  a  point  outside  the  wire  may  be  obtained  as 
follows  :  Let  P  be  the  point  and  PO  a  perpendicular  let  fall 
on  the  wire.  Take  0  as  origin  and  measure  off  any  distance  x 
(see  Fig.  1)  along  the  wire.  Let  Bx  be  an  element  of  length  at 
this  distance,  and  let  the  charge  on  the  wire  be  q  electrostatic 


0 


<fcc 


FIG.  1. 


units  per  unit  of  length  of  the  wire.     Then  the  electric  force 
due  to  the  charge  q&x  on  8x  at  P  in  the  direction  PO  is 


where  r  is  the  length  PO. 

Hence  the  electric  force  at  P  due  to  the  whole  charge  on  the 
infinitely  long  wire  resolved  in  the  direction  PO  is 

of          rqSx- 


(8) 


But 


Hence 


1 


r&c 


dV 
dr 


since  the  force  F  is  the  rate  of  decrease  of  the  potential  V  at  P 
in  the  direction  of  F. 


192   PROPAGATION  OF  ELECTEIC  CUERENTS 

dV    2g 
Accordingly  we  have         ~dr==r1 

dr 
or  dV=-2q— . 

Hence,  integrating  this  last  equation,  we  have 

7=  -2grloger  +  C     .  .     (10) 

where  C  is  some  constant  of  integration.  Availing  ourselves  of 
this  expression,  we  can  obtain  approximate  expressions  for  the 
capacity  of  aerial  telegraph  and  telephone  wires. 

3;  The  Capacity  of  Overhead  Telegraph  Wires. 

— Consider  the  case  of  two  long  circular  sectioned  wires  stretched 
parallel  to  each  other  with  their  centres  at  a  distance  I)  which  is 
large  compared  with  the  diameter  of  the  wires.  If  then  this 
distance  is  sufficiently  large  to  prevent  the  charge  on  each  wire 
disturbing  the  uniformity  of  distribution  of  the  charge  on  the 
other  wire  we  may  consider  that  the  charge  on  each  wire  is 
uniformly  distributed  round  the  surface  and  equivalent  to  a 
number  of  uniformly  electrified  filaments  arranged  on  the  surface 
of  a  cylinder  parallel  to  its  axis. 

Let  one  wire  be  denoted  by  A  and  be  supposed  to  be  charged 
positively  and  the  other  wire  be  B  and  be  charged  negatively. 
Then  the  potential  at  the  centre  of  A  may  be  denoted  by  VA,  and 
bearing  in  mind  the  expression  for  the  potential  of  a  filament  at 
any  point  outside  it,  it  will  be  clear  that  this  potential  VA  is  given 

by' 

VA  =  (-2qlogr+C)-(-2qlogD+C)  .  .  (11) 
because  the  distance  of  all  the  charge  on  A  from  the  centre  of  A 
is  r  and  the  distance  of  all  the  charge  on  B  from  the  centre  of  A 
is  nearly  D. 

Similarly  the  potential  VB  at  the  centre  of  B  is 

F£=-(-2^1ogr+C)  +  (-2^1ogD  +  C)     .         .     (12) 
and  hence 

VA  -VB=±q  (log,  D  -  loge  r) =4g  log€  ^  .     (13) 

But  the  charge  per  unit  of  length  of  the  wires  is  q,  and  their 
difference  of  potential  is  VA  —  VB,  therefore  the  capacity  per 

unit  of  length  C  is.  q/(VA—VB)  =  -      — ^,  electrostatic  units. 


THE  CONSTANTS  OF  CABLES        193 

Accordingly  the  mutual  capacity  for  a  length  I  cms.  of  the  two 
wires,  each  of  diameter  d  cms.  and  distance  D  cms.,  where  D  is 
large  compared  with  d,  is  given  in  microfarads  by  the  expression 

C(inmfds.)  = 


4  x  2-3026  x  9  x  105  x  Iog10     : 

The  factor  2'  3026  is   the  multiplier  for  converting  logarithms 
to  the  base  10  to  Napierian  logarithms.     The  above  reduces  to 

„   ,     0-0000001208/ 
C(mmfds.)  =  -  .         .         .     (15) 

lo**T 

Since  1  mile  =  160934*4  cms.,  the  capacity  per  mile   of   two 
such  parallel  wires  at  a  distance  D  is 

.    .    .    .  (16) 


provided  D  is  large  compared  with  d  and  the  wires  are  both  very 
high  above  the  earth. 

If  the  wires  are  at  all  close  together  the  capacity  per  unit  of 
length  is  greater  than  that  given  by  the  above  formulae.  The 
mutual  attractions  disturb  the  uniform  perimetral  distribution  of 
the  charges,  and  the  calculation  of  the  capacity  becomes  much 
more  difficult. 

In  ordinary  overhead  telephone  wires  the  lead  and  return  will 
generally  be  sufficiently  far  apart  to  make  the  formulae  approxi- 
mately correct,  but  for  twin  wires  enclosed  in  the  same  insulating 
sheath  where  the  wires  are  not  more  than  two  or  three  diameters 
apart  the  above  formulae  are  not  sufficiently  correct  to  do  more 
than  give  an  approximation.  Moreover,  in  the  latter  case  the 
expressions  for  the  capacity  have  to  be  multiplied  by  a  factor 
called  the  dielectric  constant,  or  specific  inductive  capacity  of  the 
dielectric. 

A  derivative  case  of  the  above  is  that  of  a  single  wire  placed 
parallel  to,  and  at  a  height  h  above,  the  surface  of  the  earth. 

If  we  suppose  the  earth's  surface  to  be  a  good  conductor  and 
at  zero  potential,  then  the  difference  of  potential  between  the 
charged  wire  at  a  height  h  above  the  earth  and  the  earth  would 
be  half  of  that  between  the  charged  wire  and  a  similar  oppositely 

B.C.  o 


194        PKOPAGATION   OF   ELECTRIC   CURRENTS 

charged  wire  at  a  depth  h  below  the  surface  of  the  earth,  supposing 
all  the  earth  then  removed.  Hence  the  capacity  of  the  single 
wire  at  a  height  h  above  the  earth  must  be  double  that  of  two 
parallel  wires  at  distance  2/i  apart.  Accordingly  the  capacity  of 

a  length  I  of  telegraph  wire  parallel  to  the  earth  and  at  a  height 

07 
h  above  it  is  C  =  -     '—  ^  electrostatic  units,  where  d  is  the 

410^ 

diameter  of  the  wire. 
In  microfarads  we  have 

C(inmfds.)  =  -  -&  -         -     (17) 

2x2-3026x9xl05xlog10  j 

and  the  capacity  per  mile  in  microfarads  is  given  by 

.  .     (18) 


log,.  -3 

A  rather  more  accurate  formula  is  given  in  The  Electrician  for 
January  28th,  1910,  p.  645.     It  is 

C  (in  electrostatic  units)  =  —  ==.         .     (19) 

' 


where  r  is  the  radius  of  the  section  of  the  wire. 

4.  The  Capacity  of  Concentric  Cylinders  and 
of  Submarine  Cables.  —  The  next  important  case  is  that 
of  the  capacity  of  a  pair  of  concentric  cylinders. 

Let  us  suppose  a  conducting  cylinder  having  a  circular  cross 
section  of  radius  RI  to  be  placed  concentrically  in  the  interior  of 
a  conducting  cylinder  of  inner  radius  R%.  Let  the  inner  cylinder 
be  charged  with  positive  electricity.  Then  this  will  induce  an 
equal  negative  charge  on  the  inner  surface  of  the  outer  cylinder, 
and  we  shall  assume  that  this  outer  cylinder  is  connected  to 
earth.  These  charges  may  be  considered  to  be  made  up  of 
filamentary  charges  laid  along  the  surfaces. 

Let  the  cylinders  be  so  long  that  the  effect  of  the  end  distri- 
butions may  be  neglected,  and  let  the  charge  per  unit  of  length 
on  the  inner  or  outer  cylinder  be  q  electrostatic  units.  Then, 
since  all  the  filamentary  charges  are  at  the  same  distance  from 


THE  CONSTANTS  OF  CABLES        195 

the  centre,  the  potential  at  the  centre  of  the  inner  cylinder,  which 
we  shall  call  V,  is  given  by 

V=(-2q  loge  B1  +  C)  -  (  -  2tf  loge  R2  +  C), 

or  F=2glog^|  .....     (20) 

But  the  whole  charge  on  the  cylinders,  assuming  them  to  have 
a  length  I  and  supposing  the  irregularity  in  distribution  at  the 
ends  to  be  neglected,  is  ql  =  Q. 

The  capacity  per  unit  of  length  of  the  cylinders  is  then 
q/V  :=  tf,and 

C=-^         ....     (21) 
2  log,  I 

If  the  capacity  is  reckoned  in  microfarads  and  ordinary 
logarithms  used  we  have 

C  (in  mfds.)  =  -  -  -  £—  •  -  •         •     (22) 

2  x  2-3026  xlog10E2x9xlO 

If  the  dielectric  used  between  the  cylinders  has  a  dielectric 
constant  K,  then  the  capacity  for  a  length  I  is 

TCI 

C  (in  mfds.)--  -g-      -         •     (23) 

4-6052x9xl05xlog10  -^ 

Since  1  mile  =  160934'4  cms.,  and  since  the  constant 

1609344 
?6052x9xlOs== 
we  have  for  the  capacity  per  mile  the  expression 

0-0388^ 
C  (in  mfds.)=—         p  .         .         .     (24) 


where  K  is  the  dielectric  constant.  For  gutta-percha  K  =  2*46, 
for  india-rubber  (pure)  K  —  2*12,  for  india-rubber  (vulcanised) 
K  =  2*69,  and  for  paper  insulation  K  =  about  1'25  or  less. 

5.    Formulae   for   the    Inductance    of  Cables. 

The  inductance  of  a  circuit  is  that  quality  of  it  in  virtue  of 
which  energy  is  associated  with  the  circuit  when  a  current 
exists  in  it.  It  is  defined  numerically  by  the  total  magnetic 
flux  or  total  number  of  lines  of  magnetic  flux  which  are  linked 

o  2 


196        PROPAGATION   OF   ELECTRIC   CURRENTS 

with  the  circuit  when  unit  current  flows  in  it  and  when  no 
other  currents  or  magnetic  fields  are  in  its  neighbourhood. 
The  creation  of  the  magnetic  field  embracing  a  circuit  when 
an  electric  current  is  started  in  it,  requires  the  expenditure  of 
energy,  and  as  long  as  it  exists  it  represents  a  store  of  energy. 

This  energy  is  measured  by  ^Li2,  where  i  is  the  current  and 
L  is  the  inductance  of  the  circuit.  This  is  proved  in  the 
following  manner  : 

If  an  electromotive  force  v  is  applied  to  a  circuit  and  creates 
in  it  a  current  i,  and  if  this  state  of  affairs  endures  for  a 
small  time  dt,  then  the  work  done  on  the  circuit  is  vi  dt.  If 
the  circuit  has  a  resistance  72  the  energy  dissipated  in  it  by 
resistance  is  Ri2dt,  and  hence  the  difference  (vi  —  Ri2)dt  must 
represent  the  energy  stored  up  in  connection  with  the  circuit 
in  the  time  dt.  The  expression  may  be  written  (v  —  Ri)idt, 
and  therefore  v  —  Ri  must  be  a  counter-electromotive  force 
created  in  the  circuit  as  the  current  increases  in  it.  By 
Faraday's  law  of  induction  the  electromotive  force  must  be 
measured  by  the  time  rate  of  increase  of  the  total  self-linked 
magnetic  flux.  Let  L  be  the  inductance  of  the  circuit  ;  then  Li 
is  the  self-linked  magnetic  flux  when  a  current  i  exists  in  the 

circuit,  and  therefore  L-^  must    be    the   counter-electromotive 

force  due  to  the  variation  of  this  self-linked  flux.  Accordingly 
we  have  the  equation 

di 


or 

Ljt+Ri  =  v          ....     (25) 

as  the  differential  equation  connecting  the  current  in  the  circuit  i 
with  the  impressed  electromotive  force  v  at  any  instant. 

Also  the  energy  stored  up  in  connection  with  the  circuit  in 

di 
a   time   dt  must  be  L-^  i  dt  =  Li  di,  and  in  establishing  a 

current  which  starts  from  zero  and  reaches  a  final  value  I  the 
total  energy  stored  up  must  be  equal  to 


THE  CONSTANTS  OF  CABLES        197 

If  L  is  a  certain  coefficient  or  number  called  the  inductance 
of  the  circuit,  then  when  a  current  i  flows  in  the  circuit  the 
total  magnetic  flux  produced  which  is  self-linked  with  the  circuit 
is  measured  by  Li.  The  total  energy  associated  with  the  circuit 
is  measured  by  JLi'2,  and  the  counter-electromotive  force  due 

to  the  variation  of  this  self-linked  flux  is  measured  by  L  -i  - 

The  quantity  L,  or  the  inductance,  is  measured  in  terms  of  a 
unit  called  one  henry,  and  since  the  dimensions  of  this  quantity 
in  electromagnetic  measure  are  those  of  a  length,  the  absolute 
electromagnetic  measurement  of  inductance  is  expressed  in 
centimetres.  The  calculation  of  the  inductance  of  a  circuit  is 
effected  by  ascertaining  the  potential  energy  associated  with  two 


FIG.  2. 

similar  circuits  when  unit  current  flows  in  each,  and  the  circuits 
are  placed  parallel  and  at  a  certain  distance  apart.  This  may  be 
accomplished  by  means  of  a  formula  due  to  Neumann,  the  proof 
of  which  is  to  be  found  in  many  advanced  text-books  on  electrical 
theory.  It  is  as  follows  :  Let  ds  and  ds'  be  elements  of  length, 
one  in  each  of  the  two  circuits,  and  let  6  be  the  angle  between 
their  direction,  and  r  the  distance  between  them.  Then  the 
mutual  inductance  M  of  the  two  circuits  can  be  found  by  taking 

the  integral 

,.     ffCos  0  7    ,  , 

M=  II  — —dsds'     ....     (26) 

where  the  integration  is  extended  to  every  possible  pair  of 
elements. 

Suppose,   for   instance,  we   consider  two  very  thin,  straight 
parallel  wires  of  length  I  placed  at  a  distance  b  apart.     Then, 


198       PROPAGATION   OF   ELECTRIC   CURRENTS 

taking  the  origin  at  the  end  of  each  wire,  we  define  one  element,  dx, 
in  one  wire  by  its  distance  x  from  the  origin,  and  the  other 
element,  dy,  by  its  distance  y  from  the  other  origin.  The  distance 
apart  of  these  elements  is  V(x-yy2+b*9  and  their  inclination  is 
zero.  Hence  Cos  0  —  1  (see  Fig.  2). 
The  mutual  induction  is  then  given  by 

(27) 


The  integral       -==p= 

(l        dx  ,  ^ 

and  hence          -_  —  _-—  =  log 


Again,  {(Z-2/)  +  V(l-y)*+b*}  dy 


we  can  write      M=^og  _     ^qrp+6  .     (33) 

and  if  6  is  small  compared  with  Z  this  reduces  to 

lf=:2/{log|-l}      ....     (34) 

or  M=%1  log  2^-2^-2?  log  6. 

Therefore  the  expression  for  M  if  £  is  constant  and  b  varies  is  of 

the  form 

M=A-Blogb      ....     (35) 

where  A  and  B  are  constants,  and  the  logarithms  are  Napierian. 

The  above  formulae  apply  to  the  case  of  a  pair  of  infinitely  thin 

or  filamentary  currents.     In  the  case  of  actual  conductors  we 


THE   CONSTANTS   OF   CABLES  199 

have  the  current  distributed  over  a  finite  area  or  circumference. 
We  may  either  have  the  current  uniformly  distributed  over  the 
cross  section  of  the  conductor,  as  in  the  case  of  steady  or  of  low 
frequency  currents,  or  we  may  have  it  distributed  over  the  surface 
of  the  conductor  or  round  the  periphery,  as  in  the  case  of  high 
frequency  currents.  If  then  we  deal  with  a  pair  of  parallel  wires 
of  finite  section  we  must  consider  the  actual  current  as  made  up 
of  filamentary  currents  either  laid  round  the  circumference  of 
the  wire  or  closely  packed  together  uniformly  over  the  cross 
section.  In  any  case  we  shall  have  to  obtain  the  actual  mutual 
inductance  by  taking  the  mean  value  of  a  number  of  expressions 
such  as  M  —  A  +  B  log  b,  where  the  b  applies  to  the  perpen- 
dicular distance  of  a  pair  of  selected  filaments,  one  in  one  wire 
and  the  other  in  the  other  wire.  The  final  result  will  be  that  in 
place  of  b  we  shall  have  a  certain  distance  E  such  that  log  E  is 
the  mean  value  of  all  the  values  of  log  b  for  all  possible  pairs  of 
filaments.  If 

log  R=  -n  (log  &!+log  &2+log  6,+etc.), 

71 

1 

then  fisfyt.^.^,)*  .     (36) 

and  E  is  called  the  geometric  mean  of  bi,  62,  b3,  etc. 

Hence  the  mutual  inductance  of  two  wires  of  finite  section  and 
length  I  is  given  by  the  expression 

97 

g-J-l|     .  .     (37) 

where  E  is  the  geometric  mean  distance  (G.M.D.)  of  all  possible 
filamentary  elements  into  which  we  can  divide  the  currents,  one 
being  taken  in  one  wire  and  one  in  the  other. 

The  determination  of  this  G.M.D.  is  a  purely  mathematical 
operation,  and  it  can  be  shown  that  if  the  current  is  distributed 
over  the  surface  of  a  circular-sectioned  wire,  as  it  is  in  the  case 
of  very  high  frequency  currents,  we  have  to  find  the  G.M.D.  of 
all  possible  pairs  of  elements,  in  the  circumference  of  two  circles, 
whilst  if  the  current  is  a  low  frequency  or  continuous  current  we 
have  to  find  the  G.M.D.  of  all  elements  of  area  in  the  cross 
section  of  the  two  wires,  one  element  being  taken  in  or  on  each 
wire. 

By  the  self-induction  or  inductance  of  a  circuit  we  mean  the 


200        PROPAGATION   OF   ELECTRIC   CURRENTS 

inductance  of  the  circuit  on  itself  or  the  total  flux  per  unit  of 
current  which  is  self-linked  with  the  circuit.  Hence  to  calculate 
the  inductance  of  a  straight  wire  we  apply  the  above  formula, 
but  the  quantity  E  becomes  the  G.M.D.  of  all  the  elements  of 
current  in  that  conductor  itself. 

If  the  current  is  a  high  frequency  current  or  confined  to  the 
surface,  say,  of  a  circular-  sectioned  wire,  we  have  then  to  find 
the  G.M.D.  of  all  possible  pairs  of  points  on  the  circumference 
of  a  circle,  and  Maxwell  has  shown  that  if  d  is  the  diameter  of 
this  circle,  then  the  G.M.D.  of  all  pairs  of  elements  of  the 

circumference  is  -~.1 

If,  however,  the  current  is  a  direct  or  low  frequency  current, 
then  we  have  to  find  the  G.M.D.  of  all  possible  elements  of  the 
cross-sectional  area  ;  and  if  the  cross  section  is  a  circle,  Maxwell 

has  shown  that  this  G.M.D.  is  equal  to  2^*=  2  X  °'7788> 
where  e  is  the  base  of  the  Napierian  logarithms.  Hence  if  we 
have  a  single  straight  wire  of  circular  section,  diameter  d  and 
length  Z,  its  inductance  L  is  found  by  substituting  in  the  formula 


for  the  value  of  b  either  b  =  -^oic  b  =  ^  t4  according    as    the 

current  is  assumed  to  be  distributed  over  the  surface  only  or 
over  the  whole  cross  section. 

For  the  kind  of  wires  and  for  the  frequencies  with  which  we 
are  concerned  in  telegraphy  we  may  generally  assume  that  the 
current  is  distributed  uniformly  over  the  cross  section  of  a 

circular  wire,  and  hence,  putting  b  =     e   ,  we  have 


.        .        .        .     (38) 
as  the  expression  for  the  inductance  of  a  wire  of  diameter  d  and 

o 

length    1.      For    high   frequency   currents    the    constant  -j-  is 
replaced  by  1. 

1  See  Maxwell,   "  Treatise  on  Electricity  and   Magnetism,"  2nd  Ed.,  Vol.  II., 
p.  298,  §  691. 


THE    CONSTANTS   OF   CABLES  201 

The  above  formula  (38)  enables  us  to  calculate  the  inductance 
per  unit  of  length  of  an  overhead  telephone  wire  provided  it  is 
made  of  non-magnetic  material  and  is  sufficiently  far  removed 
from  all  other  wires. 

It  cannot,  however,  be  applied  to  a  wire  made  of  iron  or  to  a 
submarine  telegraph  cable  in  which  a  single  stranded  insulated 
copper  wire  is  enclosed  in  steel  armour,  since  in  these  cases  the 
magnetic  permeability  of  the  iron  increases  the  inductance  by  a 
certain  unknown  amount  very  difficult  to  predict. 

In  the  case  of  a  pair  of  parallel  wires,  if  the  wires  are  not  so 
near  that  the  distribution  of  current  over  the  cross  section 
of  the  wires  is  disturbed  or  if  the  wires  are  very  thin  we  can 
calculate  the  inductance  as  follows  :  If  one  of  these  wires  is  a 
lead  and  the  other  a  return,  then  their  inductance  is  defined  to 
be  the  magnetic  flux  per  unit  of  current  which  is  self-linked 
with  this  circuit.  It  is  therefore  equal  to  twice  the  difference 
between  the  mutual  induction  of  the  two  wires  when  close 
together  and  when  separated  by  a  distance  D. 

If  we  consider  a  circular-sectioned  wire  of  diameter  d  to  have 
a  filamentary  conductor  placed  close  to  it  and  therefore  at  a 

mean  distance  ^  the  mutual  inductance  is  equal  to  A  —  Zl  log  H. 

If  then  the  filament  is  removed  to  a   distance  D  the  mutual 
inductance  is  equal  to  A  —  %l  log  D. 

Accordingly  the  self-induction  or  inductance  is  equal  to  twice 

2D 
the  difference,  or  to  4.1  log  —r-  . 

The  formula  holds  good  approximately  for  a  pair  of  wires  of 
small  diameter  parallel  to  each  other.  Hence 


9D 

or  £=9-2104nog10  -f-     .  .     (39) 

gives  us  a  rough  expression  for  the  inductance  of  a  length  I  of  a 
pair  of  parallel  wires  each  of  diameter  d  with  their  axes  separated 
by  a  distance  D.  All  lengths  must  be  measured  in  centimetres, 
and  the  inductance  is  then  in  centimetres,  and  must  be  divided 
by  109  to  reduce  it  to  henrys.  An  expression  for  the  inductance 


202   PROPAGATION  OF  ELECTRIC  CURRENTS 

of  a  concentric  cable  is  sometimes  required.  Let  us  suppose 
that  two  conducting  tubes  are  placed  concentrically,  and  that  the 
space  between  the  two  is  filled  with  some  dielectric.  If  the 
tubes  are  made  of  non-magnetic  material,  and  if  RI  and  R%  are 
the  radii  of  the  inside  and  outside  of  the  inner  tube  and  R3  and  /?4 
are  the  inner  and  outer  radii  of  the  outer  tube,  then  Lord 
Rayleigh  has  shown  that  the  inductance  per  unit  of  length  of 
such  a  conductor  is  given  by  the  expression 

0,^8,  2  f£22-3£2  BS  JRS 

1  los  +^     ~~     +       log 


The  logarithms  are  Napierian. 

If  the  inner  conductor  is  a  solid  rod  of  radius  7?2,  then  RI  is 
zero,  and  the  expression  becomes  somewhat  simplified,  since 

7?  1 

then  the  first  two  terms  become  2  log  -^  +  ^  an^  the  third  term 
comes  in  as  a  correcting  factor. 

6.  The  Practical  Measurement  of  the  Capacity 
of  Telegraph  and  Telephone  Cables.—  We  shall  not 
attempt  to  discuss  all  the  various  methods  which  have  been 
proposed  or  used  for  measuring  the  capacity  of  cables.  The 
difficulties  with  which  this  measurement  is  attended  depend 
chiefly  upon  the  fact  that  when  an  electric  force  is  applied  to  a 
dielectric  the  displacement  which  takes  place  is  not  merely  a 
function  of  the  force  and  nature  of  the  dielectric,  but  also  of  the 
time  of  application  of  the  force  and  its  mode  of  variation.  Thus 
if  the  electric  force  is  applied  and  kept  steadily  applied  the 
displacement  increases  very  rapidly  at  first  and  afterwards 
moves  slowly,  and  even  after  a  long  time  there  is  a  slow  increase 
in  the  displacement,  which  may  be  only  a  true  dielectric  current 
or  may  be  a  conduction  current  superimposed  on  the  dielectric 
current. 

The  conduction  current  is,  however,  distinguished  from  the 
dielectric  current  by  the  fact  that  the  energy  absorbed  in 
creating  it  is  dissipated  as  heat  in  the  dielectric  and  is  not 
recoverable,  whilst  the  energy  taken  up  in  producing  the  true 


THE    CONSTANTS   OF   CABLES  203 

dielectric  current  is  recovered  ia  the  discharge  current  when 
the  condenser  is  short-circuited. 

Nevertheless  there  is  a  considerable  difference  between  the 
instantaneous  or  the  high  frequency  capacity  of  a  condenser  and 
its  capacity  with  steady  unidirectional  electric  force  applied 
continuously.  The  latter  is  considerably  larger  than  the  former 
for  some  dielectrics. 

In  the  case  of  telephone  cables  the  capacity  with  which  we  are 
concerned  is  that  which  corresponds  to  a  frequency  n  of  the 
electric  force  of  about  800  or  750,  or  say  for  which  '2-nn  =  5,000. 

In  the  case  of  submarine  cables  or  low  frequency  alternating 
current  power  supply  we  may  consider  that  the  steady  capacity 
is  the  more  important. 

Full  discussion  will  be  found  in  good  text-books  on  electrical 
measurements  concerning  the  various  methods  of  measuring  the 
capacity  of  cables  with  steady  or  low  frequency  alternating 
electric  force.  We  shall  here  only  refer  to  one  method  which 
enables  us  to  measure  the  capacity  of  a  cable  for  telephonic 
frequencies  if  necessary. 

This  method  is  that  known  as  the  commutator  method.  The 
length  of  cable  to  be  tested  is  charged  with  a  battery 
of  a  certain  electromotive  force  and  then  discharged  through 
a  galvanometer.  This  process  is  repeated  one  hundred  or  several 
hundred  times  per  second  by  means  of  a  revolving  commutator, 
and  the  successive  discharges  are  sent  through  a  galvanometer. 
This  practically  constitutes  a  continuous  current  the  value  of 
which  in  fractions  of  an  ampere  can  be  ascertained  by  employing 
the  same  battery  or  voltage  to  reproduce  the  same  deflection 
on  the  galvanometer  when  a  known  resistance  is  placed  in  series 
with  it. 

The  details  of  the  commutator  will  be  found  described  in 
other  books  by  the  author,  so  that  it  is  unnecessary  to  repeat 
them  here.1  Suffice  it  to  say  that  the  arrangements  are  such 

1  See  J.  A.  Fleming,  "A  Handbook  for  the  Electrical  Laboratory  and  Testing 
Room,"  Vol.  II.,  p.  202,  The  Electrician  Printing  and  Publishing  Company,  Ld., 
1.  Salisbury  Court,  Fleet  Street,  London,  also  "The  Principles  of  Electric  Wave 
Telegraphy  and  Telephony,"  2nd  Ed.,  p.  170,  and  "An  Elementary  Manual  of 
Kadiotelegraphy  and  Radiotelephony,"  p.  279,  both  the  latter  published  by  Messrs. 
Longmans,  Green  &  Co.,  39,  Paternoster  U<>\\.  London. 


204        PROPAGATION   OF   ELECTRIC   CURRENTS 

that  the  cable  or  capacity  to  be  determined  is  charged  and 
discharged  a  known  number  of  times  per  second  through  a 
galvanometer  by  a  known  voltage. 

One  terminal  of  the  galvanometer  and  one  of  the  battery  are 
connected  together  and  to  the  earth  or  to  one  of  the  twin  con- 
ductors or  the  outside  sheath  of  the  cable  to  be  tested,  and  the 
other  conductor  is  connected  to  the  middle  terminal  of  the 
commutator,  the  remaining  battery  and  galvanometer  connection 
being  made  to  the  two  outer  terminals  of  the  commutator. 

If  there  are  N  commutations  per  second  and  if  the  charging 
voltage  is  V  and  the  capacity  is  C  microfarads,  then  the 
current  through  the  galvanometer  is  A7CT/106.  If  this  same 
deflection  is  restored  when  the  voltage  V  is  applied  to  the 
galvanometer  through  a  resistance  E  which  includes  that  of  the 
galvanometer  itself,  then  we  must  have 
NCV  V  IGft 


Hence  the  capacity  is  measured  in  microfarads  by  the  reciprocal 
of  the  product  of  the  total  resistance  in  megohms  and  the  frequency 
or  number  of  discharges  per  second. 

This  method  has  the  advantage  that  by  employing  a  commu- 
tator running  at  a  suitable  speed  we  can  determine  the  capacity 
corresponding  to  any  required  frequency  within  limits. 

The  method,  however,  does  not  separate  out  the  true  dielectric 
current  from  any  conduction  current  unless  certain  precautions 
are  taken.  It  is  always  desirable  to  make  two  sets  of  measure- 
ments, one  with  the  galvanometer  arranged  so  as  to  measure 
the  series  of  charges  given  to  the  condenser  and  one  in  which  it 
is  arranged  to  measure  the  discharge  current.  If  these  two  sets 
of  measurements  give  different  results  the  condenser  has  leakage 
as  well  as  capacity. 

Certain  types  of  gutta-percha-covered  wire  or  cable  are  known 
to  be  characterised  by  considerable  true  leakance  as  well  as 
capacity.  That  is,  the  gutta-percha  as  a  dielectric  has  a  true 
conductivity,  perhaps  owing  to  moisture  present  in  it,  as  well  as 
dielectric  quality.  Hence  many  of  the  methods  proposed  for 
measuring  capacity  do  not  give  correct  results  in  the  case  of 
gutta-percha-covered  wire  or  cable, 


THE   CONSTANTS   OF   CABLES 


205 


By  any  of  the  ordinary  methods  of  measuring  capacity  it  is 
difficult,  if  not  impossible,  to  separate  out  the  true  conduction 
current  from  the  true  dielectric  current.  They  can,  however,  be 
distinguished  as  follows  : 

If  an  alternating  current  is  employed  to  send  a  current  through 


FIG.  3. — General  view  of  Dr.  Sumpner's  Wattmeter, 
a  condenser  the  part  of  that  current  which  depends  upon  capacity 
is  expressed  by  Cjr,  and  if  the  potential  difference  of  the  plates, 

viz.  r,  is  a  simple  sine  function  of  the  time  of  the  form  v  =  V  Sin  pt, 
then  the  capacity  current  is  measured  by  CpV  Cos  pt,  and  is 
in  quadrature  as  regards  phase  with  the  potential  difference.  If, 
however,  the  condenser  possesses  any  true  conductivity  S,  then 
the  conduction  current  is  Sv  or  SV  Sin  pt,  and  this  current  is  in 
step  with  the  condenser  potential  difference. 


206        PROPAGATION   OF   ELECTRIC   CUKKENTS 

Accordingly  we  can  separate  out  these  two  components  by 
any  method  which  takes  account  only  of  the  component  in 
quadrature  with  the  potential  difference. 

This  is  achieved  by  the  use  of  Dr.  Sumpner's  iron-cored  watt- 
meter.1 This  wattmeter,  the  general  appearance  of  which 
is  shown  in  Fig.  3,  consists  of  a  specially  shaped  laminated 
iron  electromagnet  (I)  as  in  Fig.  4,  wound  over  with  a 
very  thick  copper  wire.  If  this  winding  is  connected  to  an 
alternating  current  circuit  the  impressed  electromotive  force  is 
almost  wholly  expended  in  overcoming  the  reactance  of  the 
circuit,  since  the  resistance  is  negligible.  Accordingly  if  the 
instantaneous  value  of  this  impressed  voltage  is  v,  and  if  the 


FIG.  4. — Arrangement  of  Circuits  in  Dr.  Sumpner's 
Wattmeter. 

corresponding  total  flux  in  the  air  gap  of  the  electromagnet    is 
represented   by   b,  then,  in  accordance  with  Faraday's  law,  we 

,  ^db 

have  v=  -N-fi] 

where  N  is  the  number  of  windings  on  the  core  of  the  electro- 
magnet. 

If  then  v  varies  in  accordance  with  a  simple  sine  law  the 
magnetic  flux  must  differ  90°  in  phase  with  it.  In  the  narrow 
gap  of  this  electromagnet  a  coil  of  wire  can  swing,  and  when  a 
current  i  passes  through  this  wire  a  force  the  mean  value  of 

1  See  Dr.  W.  E.  Sumpner,  "New  Alternate  Current  Insiruments,"  Jour.  Inst. 
Elec.  Eng.,  Vol.  XLI.,  p.  237,  1908. 


THE   CONSTANTS   OF    CABLES 


207 


which  is  ib  is  excited  causing  the  coil  to  move  across  the  lines  of 
flux.  This  is  resisted  by  the  torsion  of  a  spring,  and  hence  the 
deflection  of  the  coil  becomes  a  measure  of  the  mean  value  of 
the  product  of  the  magnetic  flux  in  the  gap  and  the  current  i  in 
the  coil.  Suppose  then  that  this  current  is  the  current  through 
a  condenser  which  is  placed  in  series  with  the  coil  and  connected 
across  the  same  terminals  which  supply  the  alternating  voltage  r. 
The  current  through  this  condenser,  supposed  to  have  leakance, 
consists,  as  above  shown,  of  a  component  in  step  with  the 
voltage  and  a  component  in  quadrature  with  it.  But  this 
latter  is  in  step  with  the  magnetic  field  of  the  electromagnet, 


WATTS 


FIG.  5.— Scale  of  Dr.  Sumpner's  Wattmeter. 

and  the  former  is  in  quadrature  with  the  field  as  regards  phase. 
Accordingly  it  is  only  the  true  capacity  current  which  contributes 
to  deflect  the  coil,  as  that  alone  is  in  step  with  the  magnetic 
field.  The  deflection  of  the  coil  is  proportional  to  the  mean 
product  of  ib,  and  therefore,  if  the  scale  over  which  the  indicating 
needle  moves  is  graduated,  as  shown  in  Fig.  5,  to  give  the  value 
of  this  product  by  inspection,  we  can  obtain  from  the  scale 
deflections  the  ratio  between  the  known  true  capacity  of  a  con- 
denser which  is  placed  in  series  with  the  coil  and  the  true 
capacity  of  any  other  condenser  or  cable  substituted  for  it,  and 
dielectric  leakage  causes  no  error  in  this  measurement. 

This  method  is  in  extensive  use  for  measuring  the  capacity  of 
condensers  for  telephone  work.     For  additional  information  on 


208        PROPAGATION   OF  ELECTRIC  CURRENTS 


the  measurement  of  the  capacity  of  cables  the  reader  is  referred 
to  the  author's  "  Handbook  for  the  Electrical  Laboratory  and 
Testing  Room,"  Vol.  II.,  p.  145,  and  to  a  paper  by  Mr.  J.  Elton 
Young  on  "  Capacity  Measurements  of  Long  Submarine  Cables," 
Jour.  List.  Elec.  Eng.  Lond.,  Vol.  XXVIIL,  p.  475,  1899. 

7.  The  Practical  Measurement  of  Inductance. 

—We  shall  also  not  attempt  to  mention  all  the  various  methods 
which  have  been  suggested  for  the  measurement  of  inductance, 
but  confine  ourselves  to  the  consideration  of  one  or  two  methods 

suitable  for  the  deter- 
mination of  the 
inductance  of  cables 
with  such  frequencies 
as  are  used  in  tele- 
phony. 

The  author's  ex- 
perience has  shown 
that  one  of  the  best 
of  these  is  the  method 
devised  by  Professor 
Anderson  as  modified 
by  the  author. 

In  this  method  the 


conductor  R,  L  of 
which  the  inductance 
L  is  to  be  measured 
is  inserted  in  one  arm 


PIG.  6. — Anderson-Fleming  method  of 
measuring  small  inductances. 


of  a  Wheatstone's  bridge  (see  Fig.  6).  If,  for  instance,  we  have 
to  determine  the  inductance  of  a  twin  cable,  it  can  be  short- 
circuited  at  the  far  end  and  the  two  home  ends  joined  into 
the  bridge  arm.  If  it  is  a  single  wire,  such  as  an  over- 
head telephone  wire,  then  a  loop  of  some  kind  must  be  formed 
enclosing  a  sufficiently  large  area  so  that  the  inductance  is 
practically  equal  to  that  of  a  straight  wire  with  the  return  far 
removed.  The  same  applies  to  an  armoured  cable  like  a  sub- 
marine cable.  We  cannot  properly  determine  the  inductance  of 
such  a  single  wire  or  cable  when  coiled  in  a  tank  or  in  a  ship, 


THE  CONSTANTS  OF  CABLES        209 

because  then  the  inductance  of  the  cable  is  increased  by  the 
mutual  inductance  of  the  various  coils  or  turns. 

In  any  case,  the  conductor  having  been  joined  into  the 
bridge,  the  bridge  circuits,  P,  Q,  and  S  are  balanced  in  the 
usual  way.  The  galvanometer  must  then  have  placed  in  series 
with  it  an  adjustable  resistance  r  and  a  condenser  C  arranged 
as  in  Fig.  6.  The  battery  circuit  must  have  a  buzzer,  or 
interrupter,  K,  placed  in  it  so  as  to  interrupt  the  battery 
current  several  hundred  times  per  second.  In  place  of  the 
galvanometer  a  telephone  T  is  inserted.  The  bridge  arms 
having  been  adjusted  to  obtain  a  steady  balance,  so  that  no 
current  flows  through  the  galvanometer  when  the  buzzer 
is  short-circuited,  we  switch  over  to  the  telephone  and  replace 
the  buzzer.  A  loud  sound  will  then  be  heard  in  the  telephone, 
and  this  must  be  annulled  by  inserting  resistance  r  in  series  with 
the  telephone.  When  silence  has  been  obtained  the  inductance  L 
of  the  cable  under  test  is  given  by  the  formula  below. 

Let  the  four  resistances  forming  the  arms  of  the  bridge  be 
P,  Q,  E,  S,  R  being  the  resistance  of  that  arm  which  includes 
the  inductance  L.  Let  x  be  the  current  in  arm  Q,  and  let  z  be 
the  current  in  the  resistance  r  and  y  that  in  the  inductive 
resistance  LR. 

If  then  the  bridge  is  balanced  so  that  P  :  Q  =  R  :  S  there  will 
be  no  current  in  the  galvanometer  when  the  battery  current  is 
steady.  If  r  is  so  adjusted  that  there  is  no  current  in  the  tele- 
phone when  the  battery  current  is  interrupted,  then  the  fall  of 
potential  down  S  must  be  equal  to  the  fall  of  potential  down  Q 
and  r,  and  the  current  in  r  must  be  the  same  as  the  condenser 
current.  Also  the  fall  of  potential  down  P  must  be  the  same  as 
that  down  the  inductive  resistance  LR.  These  conditions 
expressed  in  symbols  are 


PS  =  QR,  and  ^  (zdt  =  Sy. 
From  these  equations  we  easily  find  that 


B.C. 


210        PROPAGATION   OF   ELECTRIC   CURRENTS 

Hence  L  =  C{S(r+P)+Er}, 

or  L  =  C{r(R+S)+BQ}.  .     (41) 

In  measuring  small  inductances  the  capacity  C  should  be  small. 
The  method  is  sufficiently  sensitive  to  measure  the  inductance  of 
a  few  yards  of  wire  provided  that  the  value  of  C  is  accurately 
known.  If  the  inductive  resistance  has  iron  involved  in  its  con- 
struction, then  the  inductance  will  vary  with  the  current  through 
it  unless  that  current  is  either  very  large  or  very  small.  For 
the  purposes  of  this  test  it  is  a  great  convenience  to  have  a  small 
alternator  giving  an  electromotive  force  which  can  be  varied 
by  the  excitation  and  a  frequency  which  is  between  500  and 
1,000.  We  can  then  determine  the  inductance  for  telephonic 
frequencies. 

8.  The  Measurement  of  Small  Alternating 
and  Direct  Currents. — The  small  alternating  or  periodic 
currents  with  which  we  are  concerned  in  telephony  are  best 
measured  by  means  of  some  form  of  thermoelectric  ammeter. 
The  ordinary  telephonic  current  is  a  current  of  a  few  milliamperes 
created  by  an  electromotive  force  of  2  to  10  volts,  and  is  of 
complex  wave  form. 

According  to  Mr.  B.  S.  Cohen,  the  frequency  of  the  fundamental 
harmonic  lies  generally  between  100  and  300,  and  that  of  the 
highest  harmonic  between  4,000  and  5,000,  although  harmonics 
above  1,500  are  comparatively  unimportant.1 

The  average  frequency  of  the  telephone  speech  current  is  about 
800.  Hence  for  currents  of  such  frequency  almost  the  only 
reliable  method  of  current  measurement  is  by  some  form  of 
thermal  ammeter. 

Mr.  Duddell  has  devised  a  very  sensitive  thermoelectric 
ammeter  with  negligible  inductance.  The  current  to  be  mea- 
sured is  passed  through  a  small  wire  or  metallic  strip,  which  may 
be  gold-leaf,  supported  on  a  non-conducting  base.  Over  this  strip 
is  suspended  by  a  quartz  fibre  a  light  bismuth-antimony  thermo- 
couple, one  junction  of  which  nearly  touches  the  wire  or  strip. 

1  See  Mr.  B.  S.  Cohen,  "  On  the  Production  of  Small  Variable  Frequency  Alternat- 
ing Currents  suitable  for  Telephonic  and  other  Measurements,"  Phil.  Mag., 
September,  1908,  also  Proc.  Phys.  Soc,  Land.,  Vol.  XXI, 


THE   CONSTANTS   OF   CABLES 


211 


This  thermocouple  hangs  in  a  strong  magnetic  field,  and  when  a 
current  is  passed  through  the  strip  it  is  heated ;  this  heats  the 
thermoj unction  by  radiation  and  convection,  and  the  current  so 
created  causes  the  thermocouple,  which  is  in  the  form  of  a  long 
narrow  loop,  to  be  deflected.  The  deflection  is  rendered  visible 
by  a  light  mirror  attached  to  the  thermocouple,  from  which  a 
ray  of  light  is  reflected  to  a  scale.  A  general  view  of 
the  instrument  is  shown  in  Fig.  7.  It  can  be  calibrated 


FIG.  7. — Duddell's  Therm ogalvanometer. 

by  passing  known  small  continuous  currents  through  the 
heated  strip.  To  secure  good  readings  the  instrument  must 
be  placed  on  a  very  steady  support  free  from  every  trace  of 
vibration.  It  is,  however,  a  very  suitable  instrument  for  the 
measurement  of  the  root-mean-square  (R.M.S.)  values  of  such 
currents  as  are  usual  in  telegraph  and  telephone  cables.  By  the 
employment  of  suitable  heater  resistances  it  can  be  used  for 
large  alternating  currents. 

Another  useful  current-measuring  instrument  is  the  barretter 

p  2 


212        PROPAGATION   OF   ELECTRIC   CURRENTS 


of  Mr.  B.  S.  Cohen.  The  sensitive  portion  consists  of  a  pair  of 
small  carbon  filament  24-volt  glow-lamps.  When  the  carbon 
filament  is  heated  the  resistance  decreases.  The  two  glow-lamps 
are  joined  up  as  shown  in  Fig.  8.  Each  glow-lamp,  called  in  this 


Adjustable 
resistance 


FIG.  8. — Arrangement  of  Circuits  in  Cohen's  Barretter. 

connection  a  barretter,  has  a  pair  of  2-mfd.  condensers  attached 
to  its  terminals  and  a  shunt  connecting  them.  On  the  other  side 
a  few  cells  of  a  storage  battery  and  an  adjustable  resistance  and 
inductance  coil  are  connected  as  shown  in  the  diagram.  The 
batteries  can  send  current  through  the  carbon  filaments,  but  not 

through  the  con- 
densers, whilst,  on  the 
other  hand,  alter- 
nating currents  can 
pass  through  the  con- 
den  sers,  but  are 
throttled  by  the  in- 
ductance coils.  I  n 
each  alternating  cur- 
rent branch  of  each 
circuit  there  is  an 
interruption,  marked 
A  and  B  respectively. 
In  using  the  instru- 
ment the  adjustable 
resistances  are  given 
such  values  that  the  continuous  currents  balance  one  another, 
and  the  galvanometer,  G,  remains  at  zero.  Suppose  then  the 
alternator  removed,  and  that  some  circuit  in  which  there  is  a 


FIG.  9. — General  appearance  of  the  Cohen 
Barretter  as  made  by  Mr.  E.  Paul. 


THE  CONSTANTS  OF  CABLES        213 

feeble  alternating  current  is  connected  on  at  one  gap,  A.  This 
alternating  current  flows  partly  through  one  barretter  and  lowers 
the  resistance  of  the  filament,  and,  the  balance  being  upset,  the 
galvanometer  deflects.  The  instrument  may  be  calibrated  by 
sending  through  it  various  vsmall  alternating  currents,  which 
pass  also  through  a  known  inductionless  resistance.  The  drop 
in  potential  down  this-  resistance  can  be  measured  by  an  electro- 
static voltmeter,  also  previously  standardised,  and  the  measured 
fall  in  potential  gives  the  value  of  the  alternating  current,  which 
can  then.be  compared  with  the  observed  deflection  of  the  galvano- 
meter. The  process  of  calibration  is  more  difficult  than  in  the 
case  of  a  simple  thermal  ammeter,  but  when  once  carried  out  the 
barretter  can  be  used  to  determine  the  ratio  of  the  currents  at 
two  distant  points  in  a  telephone  cable,  and  hence  the  attenuation 
constant  of  the  cable.  The  general  appearance  of  the  barretter 
is  as  shown  in  Fig.  9. 

9.  The  Measurement  of  Small  Alternating 
Voltages.  The  Alternate  Current  Potentio- 
meter.— When  the  voltage  to  be  measured  is  not  very  small 
it  can  be  conveniently  determined  by  a  Dolezalek  electrometer, 
which  consists  of  a  quadrant  electrometer  of  the  Kelvin  pattern 
but  having  a  "  needle  "  made  of  silver  paper  suspended  by  a 
quartz  fibre.  The  instrument  is  used  as  an  idiostatic  electro- 
meter by  connecting  the  needle  to  one  of  the  quadrants.  If, 
however,  the  voltage  in  question  amounts  only  to  a  few  volts  or 
fractions  of  a  volt,  an  idiostatic  quadrant  electrometer  will  hardly 
be  sufficiently  sensitive.  Recourse  may  then  be  had  to  an 
alternating  current  potentiometer,  such  as  the  Drysdale-Tinsley 
form,  which  is  admirably  suited  for  many  of  the  measurements 
to  be  made  in  connection  with  cables.  This  last  instrument 
consists  of  a  standard  form  of  potentiometer  as  used  for  direct 
current  work,  but  it  is  supplemented  by  means  for  passing 
through  the  standard  wire  an  alternating  current  of  known 
value  derived  from  the  same  source  as  the  potential  to  be 
measured,  and  also  with  means  for  shifting  the  phase  of  this 
current  and  changing  its  amplitude. 

The  phase  shifting  is  accomplished  by  one  of  Dr.  Drysdale's 


214        PEOPAGATION   OF   ELECTEIC   CUERENTS 

phase-shifting  transformers  (see Fig.  10).  If  a  laminated  iron  ring 
is  wound  over  in  four  quadrants  with  coils  connected  pair  and  pair, 
and  if  these  two  pairs  are  joined  into  the  two  sides  of  a  two- 
phase  alternator  giving 
two  simple  harmonic 
voltages  differing  90° 
in  phase,  we  can  pro- 
duce thereby  a  rotating 
magnetic  field  in  the 
interior  space.  If  in 
this  space  is  placed  a 
core  wound  over  with 
one  winding  in  one 
plane,  then  if  this 
winding  is  placed  with 
its  plane  perpendicular 
to  the  field  of  one  pair 
of  coils  on  the  stator, 
an  E.M.F.  will  be  in- 
duced in  it,  and  if  the 
coil  is  turned  so  as  to 
be  perpendicular  to  the 
other  stator  field  it  will 
have  an  E.M.F.  differ- 
ing. 90°  in  phase  from 
the  former  induced  in 
it.  By  turning  this 
secondary  coil  into  any 
intermediate  position 
it  will  have  an  E.M.F. 
induced  in  it  which  has 
the  same  amplitude 
but  with  intermediate 
FIG.  10.— Drysdale  Phase  Shifting  Transformer  h  and  shifted  pro- 
as made  by  Mr.  H.  Tmsley.  ..  ,  i  ,1 

portionately     to     the 

angle  through  which  it  is  turned.  We  can  obtain  the  two  stator 
currents  in  quadrature  from  one  single-phase  alternator  by  intro- 
ducing a  shunted  condenser  into  one  circuit,  as  shown  in  Fig.  11. 


THE   CONSTANTS   OF   CABLES 


215 


Hence  the  phase-shifting  transformer  can  be  made  up  as  one 
self-contained  appliance  workable  off  any  constant  single-phase 
circuit  giving  a  simple  sine  curve  E.M.F.1 

Keturning  then  to  the  Drysdale-Tinsley  potentiometer,  we 
give  in  Fig.  12  a  perspective  view  of  the  instrument  and  in 
Fig.  13  a  diagram  of  the  connections.2  The  instrument  consists 
of  a  standard  form  of  direct  current  Tinsley's  potentiometer, 
to  which  is  added  an  electrodynamometer  or  mil-ampere  meter 
for  indicating  the  current  in  its  slide  wire.  A  phase-shifting 
transformer  can  have  its  secondary  circuit  put  in  series  with 
this  wire  by  a  throw-over  switch.  Then,  when  using  an 
alternating  current,  the 
ordinary  movable  coil 
galvanometer  is  re- 
placed by  a  vibration 
galvanometer  in  which 
the  needle  is  a  small 
piece  of  soft  iron 
suspended  by  a  wire 
in  .the  field  of  a  strong 
magnet,  which  can  be 
varied  by  a  magnetic 
shunt  (see  Fig.  14).  A 
coil  behind  the  iron 
carries  the  alternating 
current.  When  an  alternating  current  passes  through  this 
coil  the  needle  is  set  in  vibration,  and  if  the  magnetic  field 
is  varied  so  that  the  natural  time  period  of  the  vibrating  needle 
is  the  same  as  that  of  the  alternating  current,  the  amplitude  of 
motion  becomes  very  large,  and  is  observed  by  throwing  a  ray  of 
light  upon  a  mirror  attached  to  the  needle.  Means  are  provided 
for  varying  by  rheostats  the  current  in  the  slide  wire  of  the 
potentiometer.  If,  therefore,  we  desire  to  know  the  value  as 
regards  magnitude  and  phase  of  the  alternating  potential 

1  See  Dr.  C.  V.  Drysdale,  "  The  Use  of  a  Phase-shifting  Transformer  for  Wattmeter 
and  Supply  Meter  Testing,"  The  Electrician,  Dec.  llth,  Vol.  LXIL,  p.  341,  1908. 

2  See  Dr.  C.  V.  Drysdale,  "  The  Use  of  the  Potentiometer  on  Alternate  Current 
Circuits,"  Phil.  Mag.,  March,  Vol.  XVII.,  p.  402,  1909,  or  Proc.  Phy*.  An-.  Loud., 
Vol.  XXI.,  p.  561,  1909. 


Meter  or  Wattmeter 


FIG.  11. — Diagram  showing  the  manner  in 
which  two  currents  in  phase  quadrature 
can  be  obtained  from  a  single  phase 
current  by  means  of  a  shunted  condenser. 


216        PKOPAGATION   OF   ELECTRIC   CURRENTS 


THE  CONSTANTS  OF  CABLES 


217 


218        PROPAGATION   OF   ELECTRIC   CURRENTS 

difference  between  two  points  or  between  the  ends  of  a  non- 
inductive  resistance  carrying  an  alternating  current,  we  bring 
from  these  points  two  wires  to  the  potentiometer  in  the  usual 
way,  and  balance  this  unknown  alternating  potential  difference 
(A.P.D.)  against  the  fall  of  potential  (also  alternating)  down  the 
slide  wire,  and  adjust  the  strength  and  phase  of  this  fall  by  the 
rheostats  and  phase  shifter  until  the  vibration  galvanometer 
shows  no  current  (see  Fig.  15).  To  do  this  the  current  in  the 
slide  wire  must  be  provided  from  the  same  source  as  that  which 


FIG.  14. — Tmsley  Vibration  Galvanometer  for  use  with  A.  C. 
Potentiometer. 

supplies  the  current  or  potential  difference  under  test,  so  that 
the  frequency  is  the  same.  The  phase  of  the  A.P.D.  under  test 
is  then  read  off  at  once  on  the  dial  of  the  phase-shifting  trans- 
former, which  is  shown  at  the  right-hand  bottom  corner  in 
Figs.  12  and  13.  We  have  to  balance  the  A.P.D.  to  be  tested 
against  the  known  A.P.D.  between  two  points  on  a  slide  wire 
in  which  is  a  current  of  known  value,  the  phase  of  which  can 
be  shifted  if  need  be  through  360°.  The  current  in  this  wire  is 
kept  at  a  known  value  and  equal  to  that  of  a  standard  direct 
current,  which  last  can  be  adjusted  by  a  standard  Weston  cell  in 
the  usual  way. 


THE   CONSTANTS   OF   CABLES 


219 


The  instrument  forms  therefore  a  valuable  means  of  measuring 
small  alternating  currents  both  for  strength  and  phase  difference. 
We  can  by  means  of  it  determine  the  current  and  phase  of  that 


'Cvvwwwv 

Luw  Resistanc 


Load 


FIG.  15. — Scheme  of  Connections  used  in 
making  tests  with  the  Drysdale-Tinsley 
A.  C.  Potentiometer.  The  points  A,  B 
are  the  terminals  of  a  100- volt  alternator 
or  transformer. 

current  at  any  point  in  a  long  cable   to  which   an  alternating 
electromotive  force  is  applied. 


1O.  The  Measurement  of  Attenuation  Con- 
stants of  Cables.— If  the  current  at  any  point  in  a  cable 
is  Ii  and  that  at  any  other  point  separated  by  a  distance  I  is  72, 


220        PROPAGATION   OF   ELECTRIC   CURRENTS 

and   if  a  is  the  attenuation  constant   of    the    cable,   then   the 
equation  which  connects  the  above  quantities  is 


where  (7i)  and  (72)  signify  the  strengths  of  these  currents  without 
regard  to  phase  difference. 

Hence  m  =  ^    and   a=yloge^       .  .     (43) 

(J-v  *          (**) 

or,  using  ordinary  logarithms, 

a=j  2-3026  logMg      .  -     (44) 

The  attenuation  constant  a  is  therefore  quite  easily  measured 
by  inserting  in  the  run  of  the  cable  at  two  points  separated  by 
a  known  distance  I  two  hot  wire  ammeters  or  two  barretters 
which  agree  absolutely  together  and  measuring  with  them  the 
R.M.S.  value  of  the  currents  in  the  cable  at  the  two  places. 
The  attenuation  constant  is  the  Napierian  logarithm  of  the  ratio 
of  these  currents  divided  by  the  distance  in  miles  or  nauts. 

11.  Measurement   of  the   Wave    Length    Con- 
stant of  a  Cable.—  The  wave  length  constant  ft  of  a  cable 
is  defined  to  be  an  angle  ft  in  circular  measure  such  that  the 
phase  difference    in  the   currents   at   two  points   in  the   cable 
separated  by  a  distance  I  is  ftL     Accordingly  it  can  be  measured 
by  means  of  a  Drysdale-Tinsley  alternate  current  potentiometer 
or  by  any  other  means  which  enables  us  to  measure  the  phase 
difference  between  the  currents. 

12.  Measurement    of    the    Propagation   Con- 
stant  of   a    Cable.  —  The    propagation    constant    P   of    a 
cable  is  defined  by  the  equation  P=a  +  j/3,  where  a  is  the  attenua- 
tion constant  and  ft  is  the  wave  length  constant.    Accordingly  P 
is   known   when  a  and  ft  are   separately   determined.       It  is, 
however,  best  measured  by  determining  the  final  sending  end 
impedance  with  far  end  open  and  closed  as   shown  in  the  next 
section. 

13.  Measurement  of  the   Initial    Sending  End 
Impedance   of  a    Cable.—  We   have   defined    the   initial 


THE  CONSTANTS  OF  CABLES        221 

sending  end  impedance  ZQ  of  a  cable  in  Chapter  III.,  §  4,  as  the 
quantity 

VB+jpL 

z»-s%+m 

It  is  a  vector  quantity  and  is  measured  in  vector  ohms  and 
expressed  in  the  form  (X)/#,  where  (X)  is  some  number  of  ohms 
and  0  is  some  phase  angle. 

We  have  also  seen  that  the  final  sending  end  impedance  Z^  is 
defined  by  the  equation 

7      Fl 
Zi=j- 

where  V\  is  the  simple  periodic  electromotive  force  applied  to 
the  sending  end  of  a  cable  and  I\  is  the  current  flowing  into 
it  at  the  sending  end. 

Suppose  that  the  ratio  FI//I  is  measured  when  the  far  end  of 
the  cable  is  open  or  insulated  and  call  the  value  Z/,  then  we 
have  seen  (Chapter  III.)  that 

Z,=Z0CothPl        ....     (46) 

Again,  if  the  final  sending  end  impedance  is  measured  with 
the  far  end  of  the  cable  short  circuited,  and  if  we  call  this 
value  Zc,  we  have  seen  that 

Zc=:Z0T&nhPl        ....     (47) 

Hence  multiplying  together  the  equations  (46)  and  (47)  we  have 

ZQ=VzfZe (48) 

The  process  of  measuring  the  initial  sending  end  impedance 
consists  therefore  in  measuring  the  ratio  of  the  applied  voltage  V\ 
to  the  current  at  the  sending  end  when  the  receiving  end  is 
insulated  and  when  it  is  short-circuited.  It  must  be  remembered 
that  FI  and  1\  in  both  cases  are  quantities  differing  in  phase  as 
well  as  magnitude.  Hence  their  ratio  is  a  vector,  and  therefore 
the  geometric  mean  *JZfZc  is  a  vector  and  is  expressed  in  vector 
ohms. 

The  measurement  can  be  made  either  with  a  Drysdale-Tinsley 
potentiometer  or  with  a  Cohen  barretter.  It  involves  measuring 
the  value  of  I\  in  the  two  cases  and  the  difference  in  phase  of 
this  current  and  the  impressed  voltage  V\  in  the  two  cases,  but  it 


222        PEOPAGATION   OF  ELECTRIC  CUREENTS 

is  the  best  means  of  measuring  the  initial  sending  end  impe- 
dance ZQ  which  appears  in  so  many  of  the  formulae.  This 
method  of  measurement  enables  us  also  to  calculate  the  value 
of  S  +  jpC  for  any  cable,  as  the  values  of  S  and  C  are  less 
easy  to  measure  experimentally  than  those  of  R  and  L. 

Since  Zc  =  Z0  tanh  PI 
and  since  ZQ  =  \/ZfZc  it  follows  that 


?       .         .         .         .     (49) 
and  therefore  that 


.  .     (50) 

This  gives  the  best  means  of  determining  the  propagation 
constant  experimentally  in  the  case  of  any  given  cable.  Since  P 
is  an  abbreviation  for  the  product  V~R  +  jpL  VS  +  jpC  and  ZQ 
stands  for  the  quotient  Vlt  +jpL/V'S  +jpC  it  follows  that 


Hence  substituting  the  values  of  P  and  Z0  given  above  we 
have 


R+jpL=  -tanh"1  .         .         .     (51) 


The  experimental  determination  therefore  of  Zf  and  Zc  leads 
at  once  to  a  knowledge  of  the  vector  impedance  R  +jpL  and  the 
vector  admittance  S  +  jpC. 

14.  Measurement  of  the  Impedance  of  various 
Receiving  Instruments.  —  The  measurement  of  the  induc- 
tance effective  resistance  and  vector  impedance  of  various  types 
of  receiving  instrument  is  an  extremely  important  matter  because 
no  predeterminations  can  be  made  of  the  current  at  the  receiving 
end  of  a  line  unless  we  know  the  impedance  of  the  receiving 
instrument.  Some  very  valuable  measurements  of  this  kind 
have  been  carried  out  by  Mr.  B.  S.  Cohen  in  the  investigation 


THE  CONSTANTS  OF  CABLES        223 

laboratory  of  the  National  Telephone  Company  and  are  recorded 
in  the  National  Telephone  Journal1  for  September,  1909,  by 
methods  described  lower  down.  Also  other  methods  of  measure- 
ment have  been  elaborated  by  Messrs.  B.  S.  Cohen  and 
G.  M.  Shepherd  which  are  described  in  a  paper  on  Telephonic 
Transmission  Measurements  read  before  the  Institution  of 
Electrical  Engineers  of  London  in  1907,2  in  which  the  Cohen 
barretter  is  employed.  This  instrument  has  already  been 
described  in  principle  in  §  8  of  this  chapter. 
By  it  the  following  measurements  can  easily  be  made  : 

1.  The    impedance    of    any    piece    of    telephonic    apparatus 
expressed  in  ohms  for  any  type  of  alternating  current. 

2.  By  employing  an  alternator  giving  a  simple  periodic  or  sine 
form  E.M.F.  the  actual  inductance  and  effective  resistance  and 
capacity    of   any    piece  of   apparatus  for  these  high  frequency 
currents  can  be  obtained. 

3.  Small  alternating  currents  can  be  measured  with  an  ordinary 
galvanometer. 

4.  The  direct  comparison  of  various  types  of  cables  with  the 
performance  of  a  standard  cable  can  be  made. 

The  barretter  can  be  used  with  modification  to  measure  the 
impedance  of  any  piece  of  telephonic  apparatus.  For  this  pur- 
pose a  source  of  electromotive  force  must  be  provided  having 
approximately  a  simple  sine  wave  form,  and  a  frequency  of 
about  800.  Also  the  shunt  (see  Fig.  8)  must  be  replaced  by  a 
telephone  induction  coil  and  a  large  condenser  (10  mfd.)  placed 
across  the  galvanometer  terminals. 

Many  forms  of  alternator  have  been  devised  for  this  purpose, 
some  of  which  are  described  in  the  author's  work,  "  Principles  of 
Electric  Wave  Telegraphy  and  Telephony,"  Chap.  I. 

The  Western  Electric  Company  of  America  supply  a  machine 
having  an  output  of  about  30  watts  at  frequencies  varying  from 
800  to  1,800,  and  the  wave  form  is  stated  to  resemble  a  sine  curve 
closely  at  all  loads. 

Messrs.  Siemens  and  Halske  also  make  a  machine  with  an 
output  of  3  or  4  watts  with  the  same  frequencies.  This  machine 

1  Published  at  Telephone  House,  Victoria  Embankment,  London. 

2  See  Journal  of  Proc.  Inst.  Elec.  Eng.  Lond.,  Vol.  XXXIX.,  p.  503,  1907. 


224   PKOPAGATION  OF  ELECTRIC  CURRENTS 

is  of  the  inductor  type,  and  the  purity  of  the  wave  form  is  pre- 
served by  appropriately  shaping  the  teeth. 

The  investigation  department  of  the  National  Telephone 
Company  constructed  a  small  inductor  machine  giving  a  small 
output  but  approximately  sine  form  of  wave. 

For  accurate  measurements  this  machine  can  be  supplemented 
by  a  wave  filter  consisting  of  a  series  of  inductance  coils  of  low 
resistance  with  condensers  parallelised  across,  and  this  circuit  is 
so  designed  as  to  obstruct  the  passage  of  harmonics  and  preserve 
the  fundamental  sine  term  in  the  wave  form. 

Such  a  wave  filter  was  described  by  Mr.  G.  A.  Campbell  in  an 
article  in  the  Philosophical  Magazine  for  March,  1903.1 

A  fairly  good  test  of  the  simple  sine  form  of  the  E.M.F.  of 
an  alternator  is  to  employ  it  to  charge  some  form  of  condenser 
and  measure  the  charging  current.  If  this  agrees  with  that 

calculated   from    the    expression   A  =  -™-   where    C    is    the 

capacity  in  microfarads,  V  the  P.D.  of  the  condenser  terminals 
in  volts,  and  A  the  charging  current  in  amperes,  then  the  E.M.F. 
wave  form  is  very  probably  a  pure  sine  curve. 

Returning  then  to  the  actual  measurement  of  the  impedance 
of  some  form  of  telephonic  apparatus,  let  R0  be  the  effective 
resistance  of  the  apparatus.  This  must  not  be  confused  with  the 
true  steady  or  ohmic  resistance.  It  is  much  greater,  first,  because 
the  H.F.  current  in  the  conductor  is  not  uniformly  distributed 
over  the  cross  section  of  the  wire;  secondly,  because  the 
current  in  neighbouring  turns  of  wire  furthermore  increases  this 
non-uniformity ;  and  thirdly,  because  the  dissipation  of  energy 
in  any  iron  core  which  may  be  present  in  the  form  of  eddy 
currents  or  magnetic  hysteresis  loss  is  a  dissipation  of  energy 
which  counts  as  if  due  to  an  increase  in  the  actual  resistance. 

In  the  next  place  the  apparatus  has  inductance  L0,  and  at  a 
frequency  n  when  n  =  p/Zir  we  have  an  impedance  Vtt(?  +  p*L(? 
in  the  apparatus. 

Suppose  then  the  telephonic  apparatus  under  test  is  inserted 

1  See  also  Mr.  B.  S.  Cohen,  "  On  the  Production  of  Small  Variable  Frequency 
Alternating  Currents,"  Phil.  Mag.,  September,  1908,  or  Proa.  Phys.  Soc.  Lond., 
Vol.  XXI.,  p.  283,  1909. 


THE  CONSTANTS  OF  CABLES        225 

in  one  gap  B  in  the  Cohen  barretter  circuits  (see  Fig.  8)  and  a 
variable  inductionless  resistance  is  inserted  in  the  other  gap  A, 
and  let  a  high  frequency  sine  wave  alternator  be  connected  in 
as  shown  in  the  diagram. 

Let  the  barretter  or  glow  lamp  and  shunt  across  its  terminals 
together  with  the  condensers  in  series  (2  mfds.)  have  an  equivalent 
resistance  r.  The  first  step  is  to  balance  on  the  galvanometer 
any  inequality  in  the  electromotive  force  of  the  two  batteries 
inserted  in  front  of  the  barretters.  This  is  done  by  the  adjust- 
able resistances.  The  alternator  is  then  started  and  the  variable 
inductionless  resistance  RI  in  the  gap  A  is  altered  until  it  balances 
the  effect  of  the  impedance  \/Ro2  +  p2L<j2  in  the  gap  B,  and  the 
galvanometer  then  shows  no  current  because  the  effective 
impedance  in  the  circuits  of  both  barretters  is  the  same.  ' 

If  then  the  resistance  of  each  of  the  shunts  in  the  barretter 
across  the  pair  of  condensers  in  each  side  is  denoted  by  r,  and 
since  the  E.M.F.  in  two  circuits  is  the  same  and  the  currents 
the  same,  we  have  an  equality  between  the  total  resistances  or 
resistance  and  impedances,  or  in  other  words  the  equation 

.Bi+r        .         .         .     (53) 


We  need  not  take  into  account  the  reduction  in  the  shunt 
resistance  r  which  results  from  it  being  shunted  by  a  galvano- 
meter provided  the  latter  has,  as  it  should  have,  a  resistance 
of  several  thousand  ohms. 

Squaring  both  sides  of  (53)  we  have 

(B0+r)*+l>2JV  =  (A  +  r)2        •         •         •     (54) 
Hence  R02  +P2  V  =  ^2+2r  (R.-R,) 

or  VR0*+p*L0*=  A/^+2r  (R.-R,)  -     (55) 

This  gives  us  the  impedance  of  the  instrument. 

To  separate  out  the  effective  resistance  E0  from  the  reactance 
we  may  proceed  as  follows  :  Add  in  series  with  the  telephonic 
apparatus  an  inductionless  resistance  r\  and  proceed  as  before  to 
obtain  a  balance  against  an  inductionless  resistance  of  value  R% 
in  the  other  side  of  the  barretter.  Then  we  have  the  equation 

(JB0+r1+r)2+^o2  =  (^.+»-)2     •  •     (56) 

and  since  by  (54)  we  have 

(R,+r)^p^LQ^  =  (Rl  +  rY  .         .     (57) 

we  have  two  simultaneous  equations  to  determine  pL  and  R. 
B.C.  Q 


226 


Hence 


PROPAGATION   OF  ELECTRIC   CURRENTS 


.        .     (58) 
.        .    (59) 

From  which  we  obtain  tan  6  =  +-~,  0  being  the  phase  angle  of 

the  vector  impedance 


Mr.  Cohen  finds  that  the  above  method  of  measuring  the 
effective  resistance  and  inductance  of  telephonic  apparatus  can 
give  good  results  provided  that  the  shunts  shown  in  Fig.  8 


FIG.  16. — Arrangement  of  Circuits  for  measuring  the 
vector-impedance  of  any  telephonic  apparatus. 


across  the  barretter  circuits  are  replaced  by  telephone  induction 
coils  separating  the  alternator  and  gaps  A  and  B  from  the  bar- 
retter circuits,  and  also  that  a  condenser  of  large  capacity  is 
placed  across  the  galvanometer  terminals. 

Another  method  of  making  these  measurements  which  requires 
no  special  instrument  not  usually  found  in  the  laboratory 
except  the  high  frequency  alternator  was  adopted  by  Mr. 
B.  S.  Cohen  in  making  the  measurements  of  instruments  given 
below.  In  this  arrangement  the  alternator  is  applied  to  the 
battery  terminals  of  a  Wheatstone's  Bridge  (see  Fig.  16)  and  in 


THE    CONSTANTS   OF   CABLES  227 

the  bridge  circuit  is  placed  a  telephone  receiver.  The  instrument 
to  be  tested  is  placed  in  one  arm  of  the  bridge,  and  in  the 
adjacent  arm  is  inserted  a  variable  inductionless  resistance 
and  a  low  resistance  variable  inductance.  These  are  inde- 
pendently adjusted  to  give  silence  in  the  telephone  and  enable 
the  effective  resistance  E0  and  inductance  L0  to  be  separately 
equilibrated  by  resistance  plugged  out  of  the  box  and  inductance 
inserted  in  the  arm.  This  inductionless  resistance  is  made  on  a 
plan  suggested  by  Mr.  Duddell.  The  resistance  material  is  a  kind 
of  cloth  woven  with  a  silk  warp  and  fine  resistance  wire  woof  and 
has  the  property  of  possessing  extremely  small  inductance  and 
capacity,  which  is  more  than  can  be  said  for  the  ordinary  plug 
resistance  boxes  of  most  laboratories.  The  inductance  is  made 
with  two  coils,  one  outside  the  other,  the  inner  one  capable  of 
rotating  on  an  axis  so  as  to  be  turned  in  such  positions  as  to 
vary  the  mutual  inductance  of  the  two  parts  and  therefore  the 
self  inductance  of  the  two  in  series.  Turning  then  to  the 
results  obtained  by  Mr.  Cohen,  we  give  on  p.  228  a  table  published 
by  him  in  the  National  Telephone  Journal  for-  September, 
1909. 

The  figures  in  the  fourth  and  fifth  columns  give  respectively 

the  scalar  impedance  in  ohms  and  the  vectorial  angle  tan"1^- 

of  the  instrument. 

It  will  be  seen  that  the  effective  resistance  is  always  much 
greater  than  the  ohmic  or  steady  resistance.  Thus  a  so-called 
60  ohm  Bell  telephone  receiver  has  an  effective  resistance  of 
134  ohms,  an  inductance  of  18  millihenrys,  an  impedance  of 
176  ohms,  and  the  angle  of  lag  of  current  behind  terminal  P.D.  is 
40°  24'. 

The  last  column  gives  the  power  absorption  of  the  instru- 
ment in  milliwatts  per  volt  P.D.  at  the  terminals,  and  the 
total  power  loss  is  obtained  by  multiplying  these  num- 
bers by  the  square  of  the  terminal  potential  difference  in 
volts. 

We  thus  have  determined  for  us  the  value  of  the  Zr  which 
appears  in  many  formulae  in  Chapter  III.  as  the  vector 
impedance  of  the  terminal  instruments. 

Q  2 


TABLE  GIVING  THE  EFFECTIVE  EESISTANCE  B0  AND  INDUCTANCE  L0 
AND  IMPEDANCE  FOR  VARIOUS  TELEPHONIC  INSTRUMENTS  AT  A 
FREQUENCY  n  =  1000  OR  p  =  6280.  (MR.  B.  S.  COHEN.) 


Apparatus. 

S.L. 

No. 

Effective 
resistance. 
Ohms. 

Induct- 
ance. 
Henrys. 

Impedance. 

Loss  in 
milliwatts 
per  1  volt. 

Ohms. 

Angle. 

Sells. 

1,0000  magneto     . 

6 

7,580 

1-305 

11,140 

47°     9' 

•061 

Indicators. 

l,000a>  tubular,  ordinary 

10 

8,000 

1-2 

11,000 

43°  24' 

•066 

Do.         do.     differen- 

11 

20,200 

•224 

20,300 

5°    0' 

•049 

tial 

GOOw  self  -restoring 

5 

8,055 

1-3 

11,410 

44°  55' 

•062 

100«  +  100«       eyeball 

— 

3,900 

0-512 

4,035 

14°  45' 

•240 

signal,  unoperated 

100o>  +  100o>       eyeball 

— 

4,300 

0-539 

4,440 

14°    3' 

•219 

signal,  operated 

Instruments. 

Local      battery      sub- 

1 

434 

0-189 

1,265 

69°  57' 

••027 

scribers,  battery  key 

up 

Do.         do.          down 

1 

563 

0-182 

1,275 

63°  48' 

•035 

Receivers. 

Double  pole  Bell  (60a 

10 

134 

•0182 

176 

40°  24' 

4-33 

central  battery) 

Relays. 

500to  double  make  and 

9 

7,160 

1-157 

10,210 

44°  54' 

•069 

break.  (W.E.)  arma- 

ture not  attracted 

Do.        do.,       attracted 

9 

7,960 

1-238 

11,150 

44°  24' 

•064 

1,0000     do.     do.,     not 

11 

9,910 

1-543 

13,845 

44°  18' 

•052 

attracted 

Do.       do.,        attracted 

11 

9,970 

1-617 

14,230 

45°  30' 

•049 

Retards. 

lOOw  tubular      . 

— 

1,116 

0-191 

1,640 

47°    6' 

•414 

200« 

— 

3,170 

0-550 

4,690 

47°  30' 

•144 

400*         „           .         . 

5 

4,700 

0-664 

6,280 

41°  30' 

•119 

GOOo, 

1 

5,906 

0-890 

8,132 

43°  20' 

•089 

1,000»        „          differ- 

2 

19,100 

0-538 

19,400 

10°   .0' 

•051 

ential 

loca  -4-  75»    W.E.    pat- 

— 

1,827 

1-367 

8,770 

77°  58' 

•024 

tern,  No.  2020  A 

200*  +  200co        W.E. 

— 

3,600 

13-5 

85,000 

87°  34' 

•0005 

toroidal,  No.  44B 

No.     1,    Central     Battery 

Termination  (consisting 

of  repeater,  supervisory 

relay,  local  line  and  sub- 
scriber's instrument). 

(a)  No.    25  repeater, 

— 

330 

0-049 

451 

42°  57' 

1-62 

local  line,  Oo> 

(6)  Do.            do. 

— 

630 

0-068 

760 

33°  54' 

1-09 

300w  (ohmic) 

(c)  Do.           do. 

— 

680 

0-049 

746 

23°  51' 

1-22 

3-m.  20-lb.  cable 

THE  CONSTANTS  OF  CABLES        229 

15.  The  Power  Absorption  of  various  Tele- 
phonic Instruments. — The  measurement  of  the  energy 
absorbed  by  telephonic  apparatus  under  working  conditions 
presents,  as  Messrs.  Cohen  and  Shepherd  remark,  considerable 
difficulty.1  This  energy  is  extremely  small,  perhaps  only  a  few 
microwatts,  and  is  always  a  variable  quantity.  The  difficulty  is 
to  find  any  instrument  which  when  inserted  in  circuit  wif/h  the 
instrument  to  be  tested  does  not  seriously  alter  the  conditions  of 
test. 

Messrs.  Cohen  and  Shepherd  have  made  a  number  of  such 
measurements,  employing  a  method  due  to  Mr.  M.  B.  Field,  as 
follows.  If  a  small  transformer  of  suitable  design  has  one  of 
its  coils  inserted  in  parallel  with  the  instrument  under  test,  and 
if  a  suitable  inductionless  resistance  is  inserted  in  series  with  the 
instrument,  we  can  draw  off  from  the  secondary  of  the  transformer 
a  current  proportional  to  the  P.I),  at  the  terminals  of  the  instru- 
ment tested,  and  from  the  terminals  of  the  inductionless 
resistance  a  current  proportional  to  the  current  in  that 
instrument.  Let  i  be  the  current  at  any  instant  in  the 
instrument  tested  and  therefore  in  the  inductionless  resist- 
ance 11  in  series  with  it.  Then  Hi  is  the  voltage  at  the 
terminals  of  this  resistance.  Let  v  be  the  potential  difference 
at  the  terminals  of  the  instrument  tested,  then  the  P. I),  at 
the  terminals  of  the  secondary  circuit  will  be  Gv  where  G  is 
some  constant. 

A  Duddell  thermo-galvanometer  having  a  heater  with  a 
resistance  of  100  ohms  was  then  arranged  with  switches  so  that 
either  the  sum  or  the  difference  of  these  two  voltages  could 
be  applied  to  send  a  current  through  a  thermo-galvanometer 
T.G. 

Let  DI  and  D2  be  the  instantaneous  values  of  the  sum  or 
differences  of  the  above  voltages,  viz., 


TU  D*-D 

Then 


1  See  Messrs.  Cohen  and  Shepherd  on  Telephonic  Transmission  Measurements, 
Journal  Imt.  Elec.  Enrj.  Land.,  Vol.  XXXIX.,  p.  521,  1907. 


230   PROPAGATION  OF  ELECTRIC  CURRENTS 

Hence  if  we  take  mean  values  throughout  a  period  and  denote 
these  by  (L>02  (D2)2,  (I7),  and  (I)  we  have 

Cos  <£    .        .        .     (60) 

where  <£  is  the  power  factor.  The  right-hand  side  of  the  above 
equation  is  the  mean  value  of  the  power  taken  up  in  the  tele- 
phonic instrument  and  (Di)2  and  (Z)2)2  will  be  proportional  to  the 
deflections  in  the  two  cases  of  the  thermo-galvanometer. 

The  above  formula  presupposes  that  the  non-inductive  resist- 
ance R  is  very  small  compared  with  the  resistance  of  the  thermo- 
galvanometer. 

The  transformer  used  by  Messrs.  Cohen  and  Shepherd  had  a 
toroidal  core  of  No.  40  S.  W.G.  iron  wire  11*5  cm.  outside  diameter 
and  5  cm.  deep,  and  a  cross  section  of  7 '89  cms.  Its  two  windings 
had  respectively  2,000  and  100  turns  and  a  transformation  ratio 
from  96*5  to  19*3  according  to  the  number  of  secondary  turns 
used. 

The  following  results  were  obtained.  In  a  test  mac.e  with 
30  miles  of  20-lb.  paper  insulated  telephone  cable  with  far  end 
open,  the  sending  end  impedance  was  found  as  follows : — At  a 
frequency  of  810  the  current  into  the  line  was  0'00658  amp.  The 
power  absorbed  by  the  line  was  0*0163  watts,  and  the  power  factor 
was  0'71.  Hence  since  the  cable  is  fairly  long  this  gives  us  the 
initial  sending  end  impedance  Z0  =  552  ohms  with  phase  angle 
44°  48'  downwards  or  ZQ  =  552  \44°  48'. 

This  is  in  fair  agreement  with  the  calculation  made  from  the 
four  cable  constants. 

The  reader  should  note  that  the  same  method  can  be  employed 
to  determine  the  final  sending  end  impedance  when  the  cable  is 
open  or  short  circuited  at  the  receiving  end.  We  have  to 
measure,  in  that  case,  the  current  into  the  cable  at  the  sending 
end  Ii,  the  applied  voltage  or  E.M.F.  Vi,  and  the  power  taken 

up  by  the  cable  W. 
ry\ 

The  ratio  TTT-  or  the  ratio  of  the  R.M.S.  value  of  the  voltage 
W 

and  current  gives  the  numerical  value  or  size  of  the  impedance 
Zi.  Also  the  ratio  of  the  true  power  taken  up  W  in  watts  to  the 
product  of  (  Vi)  and  (/i)  or  to  the  volt-amperes  gives  us  Cos  </>  or 


THE    CONSTANTS   OF   CABLES 


231 


the  power  factor.      From  which  we  have 

(Pi)  W 

Hence  Wr  =  (ZV)  and  n7.  ,r.  =  Cos  d> 


or  the  phase  angle. 
and  the  vector  final 


sending  end  impedance  Z\  =  (Zi)  [$1 

In  the  same  manner  we  can  find  Zf,  and  Zct  and  therefore  ZQ. 

For  various  receiving  instruments  the  following  results  were 
obtained  by  Messrs.  Cohen  and  Shepherd. 


Effective 

Apparatus  tested. 
Frequency  825. 

Current  in 
amperes. 

Power  in 
watts. 

Power 
Factor. 

Resist- 
ance in 
ohms. 

Induc- 
tance in 
henrys. 

Central    Battery    Ee- 

0-00695 

0-00858 

0-600 

165 

0-0425 

ceiver 

120-ohm  Eeceiver 

0-01160 

0-02200 

0-760 

165 

0-0280 

120-ohm  Receiver  and 

0-00220 

0-00139 

0-562 

227 

0-0650 

Induction  Coil 

Central    Battery    Ke- 

0-00208 

0-00149 

0-685 

320 

0-0690 

peater  with  150-ohm 

Subscriber.'  s  Line 

16.  Determination  of  the  Fundamental  Con- 
stants of  a  Cable  from  Measurements  of  the 
Final  Sending  End  Impedance.— We  have  already 
shown  in  §  13  that  by  measuring  the  final  sending  end  impe- 
dance Zi  =  FI//I  both  with  the  far  end  of  the  cable  open  and 
closed  so  as  to  obtain  Zf  and  Zc  we  can  find  the  vector  impedance 
and  admittance  K  +  jpL  and  S  +  jpC.  Since 


S+jpC  = 


These  last  quantities  are  therefore  obtained  in  the  form  of 
complex  quantities  a-\-jb  and  can  be  drawn  as  vectors. 
Hence  we  see  at  once  that  the  horizontal  steps  of  the  two  vectors 
give  us  the  values  respectively  of  R  and  S  and  the  two  vertical 
steps  the  values  of  pL  and  pC,  from  which  L  and  C  can  be 


232        PROPAGATION   OF   ELECTRIC   CURRENTS 


obtained  since  p  —  2im  is  known.  Thus  the  four  constants  of 
the  cable  can  be  obtained  by  two  measurements  made  with  the 
Cohen  barretter  or  any  other  means  which  enable  us  to 
measure  the  impedance  of  the  cable  when  open  and  when  short 
circuited  or,  which  comes  to  the  same  thing,  the  sending  end 
current  and  its  phase  difference  and  the  impressed  voltage  in 
the  two  cases. 

Thus,  for  instance,  Messrs.  Cohen  and  Shepherd  (loc.  cit.) 
measured  the  constants  for  a  10-mile  length  of  the  National 
Telephone  Company's  standard  201b.  dry  core  paper  insulated 
cable  and  for  a  10-mile  length  of  an  equivalent  artificial  cable 
at  a  frequency  of  750  as  follows : 


— 

Impedance  in  ohms. 

Far  end  open. 

Far  end  closed. 

10-mile     length     of     standard 
cable 
10-mile  artificial  cable 

495\54°  20' 

657\29°  18' 

498\51°  28' 

644\36°    6' 

From  which  it  follows  that  for  the 

L==0.00i45    0  =  0-0540    3  =  7-12x10 
£  =  0.00020    C  =  0-0624 


10-mile  length  of)  ^  = 
standard  cable  ) 

10-mile  artificial  j  B  = 
cable  ) 


-6 


In  practice  it  is  best  to  check  the  values  of  R  and  C  by  direct 
measurements.  Since,  however,  the  constants  are  mostly  required 
in  the  expressions  Vlt*  +  p2  L?  and  VS*  +  p2  C'2  these  can  be 
obtained  directly  from  the  impedance  measurements  as  single 
numbers. 


CHAPTER  VIII 

CABLE    CALCULATIONS    AND    COMPARISON    OF    THEORY    WITH 
EXPERIMENT 

1.  Necessity  for  the  Verification   of   Formulae. 

— Since  the  object  of  all  our  investigations  is  to  obtain  rules  for 
predetermining  the  performance  of  cables  and  improving  their 
action  as  conductors,  it  is  essential  to  test  the  theory  and  formulae 
at  which  we  have  arrived  by  comparing  the  predictions  of  the 
theory  with  the  actual  results  of  measurement  in  as  many  cases 
as  possible  in  order  that  we  may  obtain  confidence  in  them  as  a 
means  of  foretelling  the  results  in  those  cases  in  which  we  cannot 
check  the  measurements  because  the  cable  is  not  then  made. 
Formulas  are  of  no  use  to  the  practical  telegraph  or  telephone 
engineer  unless  they  are  reduced  to  such  a  form  that  they  can 
be  used  for  arithmetic  calculations  of  the  above  kind  by  the  aid 
of  accessible  tables. 

It  is  essential  therefore  that  the  student  in  this  subject  should 
be  shown  how  to  employ  the  formulas  which  have  been  obtained 
in  numerical  calculations,  assuming  that  the  necessary  data  and 
tables  are  available.  In  the  last  chapter  of  this  book  are  given 
sundry  data  and  references  to  published  tables  of  various  kinds. 
We  shall  proceed  then  to  give  a  certain  number  of  instances  of 
calculation  and  verification  of  formulae. 

2.  To  Calculate  the  Current  at  any  Point  in  a 
Cable    Earthed    or   Short   Circuited   at  the    Far 
End     when      a    simple     Periodic      Electromotive 
Force  is  applied  at  the  Sending  End.— The  formula 
required  for  this  purpose  is  proved  in  Chapter  III.,  §  2,  equation 
(25).  _ 

It  is  as  follows  : 

1=1,  Cosh  Px-  —Sinh  Px  .     (1) 


234        PROPAGATION   OF   ELECTEIC    CURRENTS 

where  x  is  the  distance  from  the  sending  end,  I  is  the  current 
at  this  point,  Ii  the  current  at  the  sending  end,  P  the 
propagation  constant,  such  that  P  =  a  +  j/3,  and  Z0  is  the  initial 
sending  end  or  line  impedance 

=  VB+jpL=  B+jpL 

~  VS+jpC         a+jfi" 

The  details  of  the  following  measurements  made  with  an 
artificial  cable  by  Mr.  H.  Tinsley  have  been  communicated  by 
him  to  the  author.  These  measurements  were  made  with  a 
Drysdale-Tinsley  alternate  current  potentiometer  as  described  in 
the  previous  chapter.  The  cable  was  equivalent  to  a  submarine 
cable  having  a  length  of  230  nauts  (nautical  miles).  The  total 
conductor  resistance  was  1,440  ohms  and  the  total  capacity 
72  microfarads.  The  inductance  and  leakance  were  negligible. 
Hence  for  this  cable  we  have  the  constants 

1440 
Resistance  per  naut  R  =  ^OTT  —  6*26  ohms. 


72  0*313 

Capacity  per  naut  C  =  230xlQfi  =  -^-  farads. 

An  alternating  electromotive  of  1  volt  of  sine  curve  form  was 
applied  at  one  end  of  the  cable,  the  far  end  being  earthed.  The 
frequency  of  the  E.M.F.  was  n  =  50.  Hence  p  =  Znn  =  314. 

Accordingly  Cp  =  ^  per  naut. 

Since  L  and  S  are  negligible  we  have  for  the  attenuation  and 
wave  length  constants  the  values 


a  =  p  =  ^/  ±  CpB  =  0-0175  per  naut. 

Also  the  initial  sending  end  impedance  Z0  =     ._ _ .     Hence 

VjpC 

(Z0)  =  252-8  ohms. 

The  propagation  constant  P  =  a  +  7/3. 

Hence  P  =  0*0175  +j  0:0175. 

The  sending  end  current  /i  under  an  E.M.F.  of  1  volt  was 

0-003916  ampere,  and  this  is  so  nearly  equal  to  ^^  that   it 
shows  that  I\  =  ~  nearly.     In  other  words  the  cable  is  for  all 


COMPARISON   OF   THEORY  WITH   EXPERIMENT      235 


practical  purposes  extremely  long.     Hence  the  formula  (1)  for 
the  current  may  be  written  in  this  case 
1=1,  (Cosh  Pz-Sinh  Px) 


=  /!  (Cosh  ax  —  Sinh  ax)  (Cos  fix—  j  Sin  fix)    .         .     (2) 
Accordingly  the  strength  of  the  current  at  any  distance  x  is 
/i  (Cosh  ax  —  Sinh  ax)  amperes  and  the  phase  lags  an  angle  $x 
behind  the  current  at  the  sending  end. 

If  then  we  insert  in  the  above  formula  a  =  0*0175  and 
/!  =  0-003916  and  give  x  various  values,  say  10,  20,  30,  100,  230, 
etc.,  we  shall  have  the  predetermined  values  of  the  current  in 
magnitude  and  phase.  This  has  been  done  in  the  table  below. 

TABLE  I. 

PREDETERMINATION  OF  THE  CURRENT  AT  VARIOUS  DISTANCES  IN 
NAUTS  IN  THE  TlNSLEY  ARTIFICIAL  CABLE  FOR  WHICH 
a  =  ft  =  0-0175. 


x  —  distance  in 
nauts  from 
sending  end. 

ax  =  attenua- 
tion x  distance. 

Cosh  ax  —  Sinh  ax. 

/=  current  in 
amps. 

fix  —  phase 
angle  in 
degrees. 

10 

•175 

0-8395 

0-0033 

10° 

20 

•35 

0-7047 

0-00273 

20° 

30 

•525 

0-5910 

0-00231 

30° 

40 

•70 

0-4967 

0-00194 

40° 

50 

•875 

0-42U8 

0-00166 

50° 

100 

1-75 

0-1747 

0-00068 

100° 

150 

2-625 

0-0723 

0-00028 

150° 

230 

4-025 

0-00014 

233° 

As  a  check  on  the  above  formula  the  predictions  in  the  above 
table  may  be  compared  with  Mr.  Tinsley's  actual  measurements. 
He  measured  the  current  strength,  and  phase  difference  between 
the  current  at  any  point  and  the  sending  end  current,  and  set 
them  off  in  a  vector  diagram  shown  in  Fig.  1,  in  which  the  length 
of  each  line  drawn  from  the  origin  represents  in  magnitude  and 
direction  the  strength  and  phase  of  the  current  at  the  distances 
marked  on  it.  On  comparing  these  numbers  with  those  in 


236        PKOPAGATION  OF   ELECTRIC   CURRENTS 

Table  I.  it  will  be  seen  how  nearly  they  agree.  The  formula 
therefore  may  be  regarded  as  verified  within  the  limits  of  errors 
of  experiment. 

It  may  perhaps  be  worth  while  to  explain  in  detail  how  each 
current  value  is  calculated.  Taking  say  the  distance  of  20  nauts. 
We  have  a  =  £  =  0'0175.  Hence  ax  =  (3x  —  20  X  (V0175  =0'35. 
We  look  out  in  the  Tables  of  Hyperbolic  Sines  and  Cosines 
Cosh  0-35  and  Sinh  0'35  and  find  respectively  1-0618778  and 
0-3571898.  Their  difference  is  0'7047. 

Multiplying   this    by   0*003916    amp.  we    have  0'0033  amp., 


Cable-0- 00391 6  Amps. 


FIG.  1.  —  Vector  Diagram  of  Current  at  various  distances  along  an 
Artificial  Cable. 

which  gives  us  the  current  in  the  cable  at  20  nauts.     The  phase 
angle  is  0'35  radians  or  20°.     Similarly  for  the  other  values. 

3.  To  Calculate  the  Current  at  any  Point  in  a 
Cable  having  a  Receiving  Instrument  of  Known 
Impedance  at  the  Far  End.—  In  this  calculation  the  first 
step  is  to  find  the  final  sending  end  impedance   Z\  and    final 
receiving  end  impedance  Z%  given  the  initial  sending  end  impe- 
dance ZQ  and  the  impedance  Z.r  of  the  receiving  instrument. 
From  equations  (61)  and  (62)  in  Chapter  III.,  §  5,  we  have 
Zr  Cosh  Pl+ZQ  Sinh  PI 
Z0  Cosh  Pl+Zr  Sinh  PI   ' 

l      ,         .         .     (4) 


and  =  Cosh  Pl+      Sinh  PI         ..        .          -       .     (5) 

A  ^o 

whilst  from  equation  (25)  in  Chapter  III.  we  have 

1=  Ji  Cosh  Px  -  5  Sinh  Px  (6) 


COMPAKISON   OF   THEORY  WITH   EXPERIMENT     237 

Therefore 

r_T7fCoshPa?_Sinh  Pa?)  f. 

±  —  v\\  --  ^  —  i?  •         •         •     VU 

1       *»i  ^o 

A  verification  of  these  formulae  was  made  for  the  author  by 
Mr.  B.  S.  Cohen  by  kind  permission  of  Mr.  F.  Gill  in  the 
investigation  laboratory  of  the  National  Telephone  Company. 

The  cable  employed  was  an  artificial  line  equivalent  to  a  length 
of  the  National  Telephone  Company's  standard  cable  having  the 
following  line  constants  per  mile. 

E  =  88*4  ohms  per  loop  mile.  C  =  0'055  microfarads  per 
loop  mile.  L  and  S  negligible.  The  sending  end  electromotive 
force  was  generated  by  an  alternator  of  which  the  frequency  n 
was  1000  and  hence  p  =  2im  was  6280.  Hence  since  L  and 
S  are  zero  the  attenuation  constant  a  and  wave  length 


constant  /8  were  both  equal  to          pCR  or 


-'VI 


±  x  6280  x  0-055  x  10- 6  x  88-4  =  0-123. 
2 

Therefore  the  propagation  constant 

P=za+y/3  =  0-123-fy  0-123. 
The  initial  sending  end  or  line  impedance 

=  505X45°  vector  ohms. 


v#C     V/6280  x  -055  x  10" 
Next  as  regards  the  impedance  of  the  receiving  instrument  Zr. 
This  was  measured  and  found  to  vary  with  the  current  through 
it  as  follows  : 


Current  through  receiver 
in  milliamperes. 

Impedance  Zr  in  vector  ohms  of 
receiving  instrument. 

1-0 
2-0 
4-0 
6-0 

850    /66°    40' 

900    /67°    25' 

975     /(>8°      5' 

1030    /68°     15' 

The  line  was  then  joined  up  with  an  induction  coil  and  receiver 
at  either  end,  representing  local  battery  subscribers'  instruments, 
as  in  the  diagram  in  Fig.  2.  Alternating  current  at  a  frequency 


238        PEOPAGATION   OF   ELECTRIC   CURRENTS 

of  1000  was  then  sent  through  the  line  by  means  of  one  of  the 
induction  coils  from  a  small  sine  wave  alternator.  The  current 
at  each  end  of  the  line  was  measured  by  Cohen  barretters,  each 
barretter  being  shunted  with  a  100-ohm  shunt  and  calibrated 
under  these  conditions.  The  applied  E.M.F.  (Vi)  at  the 
sending  end  of  the  line  was  measured  with  an  Ayrton-Mather 
electrostatic  voltmeter  and  found  to  be  3'02  volts  (R.M.S. 
value). 

A  line  equal  to  a  length  of  15  miles  of  the  standard  cable 
was  then  employed  and  the  currents  measured  at  the  sending 
and  receiving  ends.  The  ratio  of  the  sending  end  to  receiving  end 
current  or  Ji//2  was  found  by  measurement  to  be  5'3.  The 
received  current  /2  was  found  to  be  1'25  milliamperes. 


<P 

uuvu     , 

,ec&Lwng  end 
Impedance. 

HN 

Ccdbfa    LLTLC 

1_ 

FIG.  2. — Experimental  Cable  with  arrangements  for  measuring  the 
Terminal  Currents. 

We  may  compare    these   numbers   with   the  predictions   of 
theory. 

The  length  I  of  cable  used  was  15  miles.     Therefore 
PZ=2-625/450=l-845+.;  1-845. 

Hence  Sinh  PZ=Sinh  (1-845 +,/'  1-845), 

and  Cosh  P/=Cosh  (1-845 +y  1-845). 

Now  1-845  radians  =  105°  44'  and  the  supplement  of  this  angle 
is  74°  16'.     We  have  then  to  calculate  the  value  of 

Sinh  (1-845+;'  1-845)  =  Sinh  1-845  Cos  105°  44'  + 

j  Cosh  1-845  Sin  105°  44' 

Cosh  (1-845+;  l-845)  =  Cosh  1-845  Cos  105°  44'  + 

j  Sinh  1-845  Sin  105°  44' 


COMPARISON   OF   THEORY  WITH   EXPERIMENT     239 

Now  Sinh  1-845  =  3-0850757, 

Cosh  1-845  =  3-2431041, 
Cos  105°  44' =  --271160, 
Sin  105°  44'=     -962534. 

Hence         Sinh  (1-845 +;l-845)  =  - 0-8364  +/3-1215 

=     3-231\105° 

Also  Cosh  (l-845+yi-845)  =  -0-8792+y2-9694 

=     3-097\106°  30'. 

Therefore       Tanh  (l-845+;l-845)  =  l-043\l°  30'. 
Also  ^0 = 505\45°~  and  Zr = 860/66°  54'. 

Hence  ^=1-7/111°  54'. 


Accordingly   ^0SinhP/=505  \45°x3-231  \105° 
=  1631/60° 
=  815+;  1412. 

Zr  Cosh  P/=860  /66°  54'  x  3-097  \106°  30' 
=  2663\173°34' 
=  -2646+;298. 

Hence  Z2=~l=ZQ  Sinh  Pl+Zr  Cosh  Pl=  -1831+;1710 

^2 

=  2500\133°40'. 


Furthermore     ^0  Cosh  Pf=505  \45°  x  3-097  \106°  30' 

=  1564/61°  30' 
=  746+;1374. 
Zr  Sinh  PI  =  860/66°  54'  x  3-231  /105C 

=  2778/171°  54' 
=  -2750+;390. 

Hence          ^0  Cosh  Pl  +  Zr  Sinh  PI  =  -  2004  +;  1764 

=  2667/138°  35'. 

„       7    ZT  Cosh  Pi  +  Z0  Sinh  PZ 
[ow  ^-^°  ^0  Cosh  P/+^r  Sinh  PI 


_^xo 


2667  \138°  35' 


=  473\49°55': 


240        PKOPAGATION   OF   ELECTRIC   CUREENTS 

Accordingly  the  four  impedances  are 
^0=505\45°  =  line    impedance    or    initial    sending    end 

impedance, 

^  =  473  \49°  55'     =  final  sending  end  impedance. 
-£2=2500  \133°  40'  =  final  receiving  end  impedance, 
^.—860  /66°  54'     =  receiving  instrument  impedance. 

Now  the  impressed  or  sending  end  voltage  was  3*02  volts. 
Therefore  we  have 

V  3*02 

Il  =  ~-  =  sending  end  current  =  -r=Q  =0-0064  amp., 

£\  4  1  o 

V  3-02 

I*=-rr  =  received  current  =gggg==(M)01108  amp., 

T^iaflB*55^  ky  calculation. 

The  ratio  -F-  was  also  found  to  be  5*3  by  observation. 
A 

The  received  current  /2  =  1'208  milliamperes  by  calculation, 
and  was  found  to  be  1*25  milliamperes  by  observation. 

Hence  there  is  a  very  good  agreement  between  the  observed 
values  and  those  predicted  by  our  formulae,  which  are  thereby 
confirmed. 

An  additional  illustration  of  the  above  formulae  may  be  given 
as  follows  : 

Suppose  a  length  of  ten  miles  of  the  same  standard  cable  to 
have  a  plain  Bell  receiving  telephone  placed  across  the  receiving 
end,  we  can  then  calculate  the  current  through  this  receiver  as 
follows  : 

The  received  current  I,  = 


^  Sinh  p          Cosh  pl 

In  this  case  we  have  for  a  ten-mile  length  of  the  cable 
^0=465-j415=625\41°  45'  ohms, 

and  Zr  =  for  a  60-ohm  Bell  receiver  is  given  approximately  by 

the  formula 

Zr=  134  +j'91  =  162/34°  15'  ohms. 

We  can  then  easily  find  that 

PI  =  10  x  0-1  +jW  x  0-1  =  1  +  jl, 
and  hence          Sinh  PI  =  0-  634  +jl  -297  =  1  -445/64°, 
and  Cosh  Pl=  0-83  +  /-99  =  1-292/50°  15'. 


COMPARISON   OF   THEORY   WITH   EXPERIMENT     241 

Hence  Zr  Cosh  PI  =  20  +./207  =  209/84°  30' 

and  Z0  Sinh  PI = 833  +;341  =  900/22°  15'. 

Therefore 

Zr  Cosh  Pl+ZQ  Sinh  PI  =  853  +;548  =  1014/32°  40'. 

Vl        10 
Accordingly       a  =  jTvfZ  =  i rn  4  ~  ^  ^ niilliamperes. 

The  reader  should  notice  that  as  PI  increases,  that  is  as  the 
length  of  the  cable  increases,  the  values  of  Sinh  PI  and  Cosh  PI 
approximate.  Since  Sinh  4  is  nearly  equal  to  Cosh  4,  and  since 
a  and  p  for  the  standard  cable  are  equal  to  about  O'l,  it  follows 
that  for  cable  lengths  of  forty  miles  and  upwards  we  can  greatly 
simplify  the  formulae  by  writing  Sinh  PI  =  Cosh  PI  in  them. 
Thus  under  these  conditions  we  have  the  receiving  end  impedance 
Zz  given  by 

and  the  received  current  72  by  the  reduced  formula 

j_  _V± ,m 

2~  (ZQ+Zr)  Sinh  Pi 

and  the  sending  end  current  by 

Ii  =  ^-      •        .  •     .        .        .     (10) 
and  the  ratio  /i//2  by 

j  =     °  g    r  Sinh  PI     .  .         .     (11) 

Thus  for  forty  miles  of  standard  cable  we  have  I  =  40, 
al  =  4  =  pi,  and  Sinh  PI  =  Sinh  4  (Cos  4  +  j  Sin  4)  = 
27'3  (Cos  4  +j  Sin  4). 

Now  for  the  same  cable  and  receiving  instrument  we  have 


91 


35' 

and  Sinh  Pl  =  27-3/229°  20'. 

Hence  Z,=(Z0+Zr)  Sinh  P£= 18591 -3/200°  45' 

and  ^  =  ^=623X41°  45'. 

Hence  for  ^  =  10  /i  =  ^  ampere  and  I2  =  ampere. 

B.C. 


242        PROPAGATION   OF   ELECTRIC   CURRENTS 


As  regards  the  ratio  of  7i/72  or  of  the  sending  end  to  receiving 
end  currents,  we  have  always 

I*~  ^o 

If  PI  is  very  small,  approximating  to  zero,  because  the  length  I 


" 

'  *v 

3 

'  N> 

'' 


J      Z     3     4     56      T     S     9    30     11    12    IB   J4    15   16   11    IS    J9  20 
-MiL&s  of  CaJblt's . 

FIG.  3. — Curves  showing  the  variation  of  the  sending  end  and  receiving  end 
Currents  in  a  Telephonic  Cable  (Cohen). 

is  small,  then   Cosh  PI  =  1    Sinh  PI  —  0  and   /i//2  =  1,  as  it 
should  be. 

If  PI  is  very  large,  say,  greater  than  4,  because  I  is  large,  then 
Cosh  PI  —  Sinh  PI,  and  we  have 


By  equation  (74)  of  §  5,   Chapter  III.,  this  equation  for  the 
ratio  7i//2  generally  may  be  written 

I1=CosMP/+y) 
Ij  Cosh  y 

17 

where  y  = 


COMPARISON    OF   THEOKY  WITH  EXPERIMENT     213 

For  certain  values  of  y  and  PI  it  is  possible  for  Cosh  (PI  +  y) 
considered  as  a  vector  to  have  a  smaller  size  than  that  of  Cosh  y. 
If  y  and  P  are  kept  constant  and  I  varied,  then  for  some  values 
of  y  and  P  we  shall  have  the  ratio  /i//2  equal  to,  less  than,  and 
greater  than  unity  as  /  progressively  increases. 

This  signifies  that  the  current  at  the  receiving  end  may,  under 
certain  conditions,  be  greater  than  the  current  at  the  sending 
end.  This  takes  place  when  /  is  small,  and  increasing  from  zero. 

This  variation  in  the  ratio  of  /i//2,  or  of  the  sending  end  to 
the  receiving  end  current,  as  the  length  of  the  cable  increases,  is 
well  shown  by  the  observations,  represented  by  the  curves  in 
Fig.  3,  which  were  taken  by  Mr.  B.  S.  Cohen  in  the  Investigation 
Laboratory  of  the  National  Telephone  Company.  For  various 
lengths  of  standard  telephone  cable  and  for  the  same  receiving 
instrument  the  currents  /i  and  1%  were  measured  with  two 
barretters,  and  the  observed  values  are  represented  by  the 
firm  line  curves  for  various  lengths  of  cable.  It  will  be  seen 
that  when  the  length  of  cable  is  zero  the  two  currents  are 
identical,  as  they  should  be.  As  the  length  of  cable  increases  up 
to  about  four  miles  the  current  at  the  receiving  end  is  greater 
than  that  at  the  sending  end.  At  a  length  of  about  4*4  miles  the 
two  currents  are  again  equal.  Beyond  that  length  the  sending 
end  current  is  greater  than  the  receiving  end  current. 

4.  To  calculate  the  Voltage  at  the  Receiving 
End  of  a  Cable  when  open  or  insulated,  and 
the  Current  when  closed  or  short  circuited.— 

The  formulae  in  this  case  are 

V^VtSoohPl          ....     (15) 

1^=^- Cosech  PI  (16) 

^o 

where  l\  is  the  impressed  voltage  at  the  sending  end,  and  V* 
and  72  the  voltage  and  current  at  the  receiving  end. 

Thus  suppose  that  V\  =  10  volts,  and  that  we  have  to  deal  with 
twenty  miles  of  standard  cable  for  which  a  =  $  =  O'l  nearly. 
Then  PI  =  20  a  + /20  /3  =  2  +j2.     Then  from  the  table  we  have 
Cosh  2  =      3-76,  Sinh  2  =  3-627, 

Cos    2=  -0-416,  Sin    2-0-909, 

since  an  angle  of  two  radians  =  114°  35'  30". 

R  2 


244        PROPAGATION   OF   ELECTRIC   CURRENTS 

Hence  Cosh  Pl=  -3-76  x  -416+y3-627  x  -909 

-1-564+J3-297 
=  3-65/115°  18', 

Sinh  Pl=  -3-627 x-416+;3-76x  "909 
=  -l-51+j3-42 


Therefore  Sech  P/  =  0-273\115°  18', 

Cosech  PZ=0-266\114°  12'. 
Hence  F2= 10x0-273  =  2-73  volts. 

Then  Z«- 


For  the  standard  cable  R  =  88  ohms,  and  L  =  *001  henry, 
and  if  we  takep  =  5,000  we  havepL  =  5  and  VR2  +  p2  L2  =  88*1. 

Also  Va?  +  /32  =  0-1414,  and  therefore  i  =  0'0016/41°  45'. 

"0 

Therefore  we  have 

/!  =  10  x  '0016  x  -266  =  0-004256. 

Hence  for  an  impressed  voltage  of  10  volts  the  voltage  at  the  far 
end  is  2'73  volts  if  the  receiving  end  is  open,  and  the  current  is 
4*25  milliamperes  if  the  receiving  end  is  short-circuited. 


5.  Calculation  and  Predetermination  of 
Attenuation  Constants.—  The  predetermination  of  the 
attenuation  constant  a  of  a  given  type  of  telephone  cable  is  a 
most  important  matter,  because  it  is  the  value  of  this  quantity 
that  determines  the  speaking  qualities  of  the  cable.  The  funda- 
mental formula  for  a  is, 


V  \ 


{  ^C^+F^2)  (Sz+p^+BS-ptLc}       .     (17) 

In  this  formula  7^  must  be  given  in  ohms,  L  in  henry  s,  C  in 
farads,  and  S  in  mhos  or  the  reciprocal  of  ohms,  and  p  is  2?r 
times  the  frequency  of  the  current. 


COMPARISON   OF   THEORY  WITH  EXPERIMENT     245 

Mr.  H.  R.  Kempe  has  pointed  out  1  that  this  formula  is  not 
very  convenient  for  calculation,  because  in  the  majority  of  cases 
the  quantity  ^(R*  +  p2L2)  (S*  +  p2C~)  +  RS  is  so  nearly  equal  to 
2)2LC  that  a  large  error  may  be  made  in  taking  their  difference 
unless  each  is  worked  out  to  many  decimal  places.  Also  it  is  more 
convenient  to  have  a  formula  in  which  we  can  insert  the  value 
of  II  in  ohms,  C,  in  microfarads,  L  in  millihenrys,  and  the 
reciprocal  of  S  in  ohms;  that  is  the  insulation  resistance  per 
mile,  naut,  or  kilometre  in  ohms,  as  given  directly  by  measure- 
ments. He  has  therefore  changed  the  above  expression  for  a 
into  another  equivalent  one  as  follows  :— 

v  c*    I    zumzzzzr          2007? 

a=/  v/^+(5^)2_5^  +  Jl+o-000128V^  +  (5L>2)  .     (18) 


In  the  above  formula  p  is  taken  as  5000  and  C  is  to  be 
understood  as  the  capacity  in  microfarads,  L  as  the  inductance 
in  millihenrys,  R  as  the  copper  resistance  in  ohms,  and  r  as  the 
insulation  resistance  in  ohms,  all  per  mile  or  per  kilometre  as 
the  case  may  be. 

If  the  cable  is  a  loaded  cable  then  the  value  of  R  is  the  con- 
ductor resistance  per  mile  phis  the  effective  resistance  of  the 
loading  coils  per  mile  and  the  value  of  L  is  the  inductance  per 
mile  of  the  cable  plus  that  of  the  loading  coils  per  mile  reckoned 
in  millihenrys. 

In  the  case  of  well-constructed  loading  coils  the  effective 
resistance  is  about  6  ohms  for  every  100  millihenrys  of  inductance. 
In  the  case  of  the  cable  itself  the  inductance  will  be  about 
1  millihenry  per  mile.  For  some  types  of  dry  core  land  cable 
the  value  of  the  insulation  conductivity  8  is  so  small  that  it  can 
be  neglected.  Under  these  conditions  we  have 


.  .     (19) 

For  unloaded  cables,  and  for  a  frequency  such  that  p  =  5000, 
we  shall  generally  have  R  greater  than  pL,  or  at  least  not  very 
different  from  it. 

1  See  Appendix  X  to  a  paper  by  Major  W.  A.  J.  O'Meara,  C.M.G.,  on  "  Submarine 
Cables  for  Long  Distance  Telephone  Circuits,"  Journal  lust.  Elec.  Eny.  Lond., 
Vol.  XLVI.,  p.  309,  1911. 


246        PROPAGATION   OF   ELECTRIC   CURRENTS 

There  is  then  no  difficulty  in  finding  the  value  of 

VB'2+p2L'2  -pL 

with  a  fair  amount  of  accuracy.  If,  however,  L  is  large  as  in 
the  case  of  loaded  artificial  cables,  then,  as  we  have  already 
shown  in  Chapter  IV., 

- 


Hence  when  pL/R  is  large  and  S  =  0  we  have  the  value  of  the 
attenuation  constant  a  given  by  the  expression  (see  p.  297), 


When  8  is  not  absolutely  zero  then  a  somewhat  more  accurate 
approximation  is  given  by  the  expression 


If  the  leakance  S  can  be  neglected,  but  if  the  inductance  L  is 
small,  even  as  small  as  one  millihenry  per  mile,  it  is  preferable 
to  calculate  the  attenuation  constant  by  the  formula 


a=/v       Cp  (V^+p^-Lp)   .        .         .     (23) 
rather  than  by  the  formula 


As  an  example  of  the  difference  the  following  values  may  be 
given,  which  were  furnished  by  Mr.  A.  W.  Martin  of  the  General 
Post  Office  in  a  discussion  at  the  Physical  Society  on  a  paper  by 
Professor  J.  Perry  on  "  Telephone  Circuits."  1  The  figures 
show  that  for  the  constants  given  the  inductance  of  the  cable 
though  small  should  be  taken  into  account  in  the  calculation. 

The  value  of  L,  the  inductance  per  mile  of  various  types  of 
cable,  is  approximately  as  follows  :— 

L  =  O'OOl  henry  per  mile  for  underground  cables. 
L  =  0*0017     „  „          ,,   submarine  cables. 

L  =  0*0032  to  0*0042  for  aerial  copper  wire  lines. 

1  See  The  Electrician,  Vol.  LXIV.,  p.  880,  March  11,  1910,  for  Mr.  Martin's 
remarks,  and  Proceedings  of  the  Physical  Society,  Vol.  XX1L,  p.  252,  1910,  for 
Prof.  Perry's  paper  on  "Telephone  Circuits." 


COMPARISON   OF   THEOEY  WITH  EXPERIMENT    >247 


TABLE  II. 

TABLE    OF    ATTENUATION    CONSTANTS    (a)    CALCULATED    AND 
OBSERVED.      >  =  27m= 


Constants  of  the  Cable  per  mile. 

Attenuation 
(  'oust  ant  (a) 
calculated  by 
Kqnat  ion  (24). 

Attenuation 
Constant  (a) 
calculated  by 
Equation  (23). 

Attenuation 
(  'on  slant  (a) 
observed. 

11 

ohms. 

c 

mfds. 

L 

henry  s. 

88 

0-050 

0-001 

0-105 

0-102 

88 

0-054 

0-001 

0-109 

0-106 

0-106 

18 

0-055 

0-001 

0-050 

0-043 

0-046 

12 

0-065 

0-001 

0-044 

0-036 

0-037 

In  practice  it  is  found  that  the  value  of  S/C  is  very  far  from 
being  negligible  when  inductance  is  introduced  into  the  cable. 
Hence  leakance  acts  to  increase  attenuation.  It  is  thus  easily 
seen  that  in  the  case  of  loaded  cables  any  large  amount  of 
dielectric  conductivity  or  small  insulation  resistance  has  a  great 
effect  in  increasing  the  attenuation  constant.  Certain  dielectrics 
such  as  gutta  percha  are  well  known  to  have  a  low  dielectric 
resistance  and  hence  create  a  relatively  large  attenuation 
constant  in  cables  insulated  with  them. 

It  has  been  stated  that  this  large  value  of  S  in  the  case  of 
gutta  percha  insulated  wire  would  nullify  the  effect  of  any 
loading  by  inductance.1  This,  however,  was  disproved  by 
experiments  made  by  Major  O'Meara,  Engineer-in-Chief  to  the 
General  Post  Office,  and  described  by  him  in  a  paper  on  Sub- 
marine Cables  for  Long  Distance  Telephone  Circuits  in  the 
following  words2 :— 

"  In  order  to  settle  the  point  definitely,  it  was  decided  to  carry 
"  out  some  experiments.  The  Department  had  a  large  stock  of 
"  No.  7  gutta  percha  covered  wire  (weight  of  copper,  40  Ibs.  per 
"  mile  ;  of  gutta  percha,  50  Ibs.  per  mile  ;  resistance,  44  ohms 
"  per  loop  mile ;  electrostatic  capacity  wire  to  wire,  0*13  micro- 
"  farad  per  mile),  and  also  a  number  of  inductance  coils 
"  (inductance,  83  millihenrys ;  resistance,  13*4  ohms  at  750 

1  See  Mrlitrutn-hnixrlicZeitxchriff,  Vol.  XXIX..  I'.ms.  j>.  588. 

2  See  Journal  Institution  Electrical  Engineer*,  London,  Vol.  XLVI.,  11)11,  p.  309- 


248        PEOPAGATION   OF   ELECTEIC   CUERENTS 

"  periods  per  second),  which  had  been  used  originally  for  carry- 
"  ing  out  some  experiments  in  connection  with  the  improvement 
"  of  transmission  of  speech  in  subterranean  cables  between 
"  Liverpool  and  Manchester.  Calculations  were  made  to 
"  ascertain  the  best  disposition  of  the  coils  in  this  particular 
"  type  of  cable — although  neither  the  coils  nor  the  cable  were 
"  really  of  the  most  suitable  type — and  it  was  found  that  in 
"  order  to  provide  55  millihenrys  per  mile  they  should  be 
"  inserted  at  intervals  of  1J  miles.  A  large  number  of  speech 
"  tests  were  made  on  loaded  circuits  formed  by  means  of  the 
"  No.  7  gutta  percha  wire,  by  myself,  Messrs.  H.  Hartnell, 
"  A.  W.  Martin,  and  other  members  of  my  staff.  It  was 
"  gratifying  to  find  that  the  actual  improvement  in  transmission 
"  was  in  complete  agreement  with  the  estimates  based  on  the 
"  calculations  that  had  been  made.  (By  calculation  the  attenua- 
"  tion  was  0*0427  per  mile,  and  the  observed  result -was  0*0419 
"  per  mile.)  We  found  that  commercial  speech  was  certainly 
"practicable  on  105  miles  of  this  particular  type  of  '  coil  ' 
"  loaded  gutta  percha  wire,  and  our  doubts  as  to  the  feasibility 
"  of  the  '  non-uniform '  loading  for  submarine  cables  of  moderate 
"  length  were  set  at  rest." 

In  the  case  of  loaded  cables  the  calculation  of  the  attenuation 
constant  can  be  carried  out  by  the  aid  of  Campbell's  formula 
given  in  §  8  equation  63  of  Chapter  IV.  This  formula  is,  how- 
ever, very  troublesome  to  work  with  owing  to  the  necessity  of 
calculating  an  inverse  hyperbolic  function  that  is  the  value  of 
Cosh"1  or  Sinh"1  for  some  vector. 

If  the  loading  coils  are  placed  at  such  intervals  that  there  are 
nine  or  ten  per  wave  reckoned  by  assuming  that  the  total 
resistance  and  total  inductance  per  mile,  including  that  of  the 
cable  itself  and  of  the  loading  coils,  are  distributed  uniformly, 
and  also  assuming  a  frequency  such  that  p  =  5000,  then  if  the 
value  of  2/T//3  where  /3  is  the  wave  length  constant  is  at  least  nine 
times  the  interval  between  the  loading  coils,  we  may  assume 
that  the  attenuation  constant  a  will  be  given  sufficiently  for  all 
practical  purposes  by  a  calculation  made  in  the  usual  manner 
with  this  uniformly  distributed  resistance  and  inductance.  An 
illustration  will  make  this  clear  : 


COMPAKISON   OF   THEOEY   WITH   EXPEEIMENT     249 

A  paper  insulated  cable  had  a  resistance  per  kilometre  of 
27*96  ohms,  a  capacity  per  kilometre  of  0*07455  microfarad,  and 
an  inductance  per  kilometre  of  0*00056  henry.  Loading  coils 
each  of  15  ohms  (effective)  resistance  and  a  total  or  double 
inductance  of  0*225  henry  were  inserted  at  intervals  of  1*2  kilo- 
metres. It  is  required  to  find  the  true  attenuation  constant  for  a 
frequency  n  such  that  Zim  =  p  =  5000. 

We  have  R  =  27'96,  C  =  0'07455  X  10~6,  L  =  0'00056; 
S  =  0  and  p  =  5000. 

For  the  line  proper  the  propagation  constant  P  where 
P  =  a  +  j/3t  and  a  and  ft  are  calculated  from  the  usual 
formula?, 


is  obtained  by  inserting  in  the  above  expressions  the  values  of 
the  R,  L  and  C  for  the  line  itself.     Hence  we  obtain 
JP= 0-06867 +;  0-07589  =  0-10234/47°  51*5'. 

Now  the  coil  interval  d  =  1*2  kilometres.     Hence 

Pd  -  0-12281/47°  51*5'  =  0*082402  +;  0-091062. 
Again  for  the  line 

VR+jpL^u .74\N42°  R-4'. 
^S+jpC 

Now 
Cosh  Pd-Cosh  (0-082402  +;  0-091062) 

-Cosh  0-082402  Cos  0-091062  +;  Sinh  0-082402  Sin 0-091062 
-0-999173+;  0-007499. 
Also 

Sinh  Pd  =  0-082146  +;  0-091219  =  0*122347/47°  59*8'. 
The  loading  coil  impedance  =  Zf  =  Rr  +  jpL'  is  equal  to 
15+;  1125 -1125-1/89°  14'. 


Also  2^0=549-48\42°  8*4'. 

Hence  ^-  =  2-Q476\131°  22*4f 

and  ^rSinh  P^=0-25052\179°  22-2' 

AAQ 

=  -0-25050+;  0*0027532. 


250        PROPAGATION   OF   ELECTRIC   CURRENTS 

By  Campbell's  formula  (see  Chapter  IV.)  if  Pf  is  the  effective 
Propagation  constant  of  the  loaded  line  we  have 

Cosh  PYZ  =  Cosh  Pdr  Sinh  Pel 


Therefore  Cosh  P'd  =  0-74867  +;  0-010252. 

Therefore  PVZ^Cosh-1  {0-74867  +t/0'010252} 

By  the  formula  in  §  5,  Chapter  I.,  we  have  then 
P'd^Cosh-1  (1  -000120)  +j  Cos-1  (0-74858) 

=  0-0155+;  0-7249. 

But  d  =  1-2  kilometres.     Hence 

P'  =0-0129  +/  0-604 

=  a'-K//3' 

where  a  is  the  effective  attenuation  constant  of  the  loaded  line. 
Accordingly  a'  =  0-0129 

and  /3'  =  0-604 

2_ 
Therefore  the  wave  length  V  =  —  -  and  A/  =  10'4  kilometres. 

There  are  therefore  10'4/1'2  =  9  loading  coils  per  wave,  and  the 
spacing  is  by  Pupin's  law  sufficiently  close. 

Suppose  then  that  the  total  resistance  and  total  inductance  of 
all  the  coils  is  smoothed  out  and  added  to  that  of  the  line,  we 
shall  have  a  total  resistance  of  27*96  ohms  per  kilometre  of 
line  and  15  ohms  due  to  the  loading  coil  per  1*2  kilometre  or 
15/1*2  =  12'4  ohms  per  kilometre.  Hence  a  total  resistance 
(R")  per  kilometre  of  27'96  +  12'4  -  40'36  ohms. 

In  the  same  way  the  total  smoothed  out  inductance  L" 
per  kilometre  is  0'00056  +  0'225/1'2  =  0-18806  -  henry.  If 
then  we  calculate  the  attenuation  constant  a"  and  wave  length 
constant  /3"  for  this  smoothed  out  cable  having  a  total 
resistance  li"  =  40'36  ohms  per  kilometre  and  a  total  induct- 
ance L"  —  0*18806  henrys  per  kilometre  and  capacity 
C  =  0-07455  X  106-°  farads  per  kilometre,  using  the  formulae 


.     (25) 

.     (26) 

we  find  we  obtain  values 

«"  =  0-0128  /T  =  0-590. 

The  smoothed  out  attenuation  constant  a"  is  therefore  very 


COMPARISON   OF   THEORY  WITH  EXPERIMENT     251 

nearly  equal  to  the  effective  attenuation  constant  a'  as  calculated 
by  Campbell's  formula.  It  has  been  shown  by  Mr.  G.  A.  Campbell 
that  if  the  spacing  of  the  coils  is  such  that  there  are  fewer  than 
9  coils  per  wave,  then  the  actual  attenuation  constant  a!  of  the 
loaded  line  is  greater  than  that  predicted  by  assuming  the  total 
resistance  and  inductance  smoothed  out  (a")  in  the  following 
proportions1  :— 

For  8  coils  per  wave  a  is  greater  than  a"  by      1% 

7  9°/ 

j»      '  »  »  »  »  ^  /o 

,,    6  „  „  „  „  3% 

3  no  I 

>>  »  »  >>  '    /o 

4  1fi°/ 

>>  )>  »  >>  »  -t*J    /O 

„    3  „  „  „  „  200%  or  more. 

As  a  rule,  therefore,  in  calculating  the  attenuation  of  loaded 
lines  we  can  proceed  as  follows.  Assume  the  total  resistance  and 
inductance  of  the  line  and  the  loading  coils  to  be  smoothed  out 
and  uniformly  distributed  and  calculate  the  resulting  E,  L,  and  C 
per  mile  or  per  kilometre  of  line.  Then  find  the  wave  length 
constant  /3  and  the  wave  length  A.  =  2?r//3  for  the  highest 
frequency  to  be  used  in  practice  or  for  the  average  frequency 
(800)  of  the  speaking  voice.  If  this  wave  length  A  is  more  than 
eight  or  nine  times  the  distance  between  the  loading  coils,  then 
we  may  proceed  to  calculate  the  attenuation  constant  with  this 
smoothed  out  resistance  and  inductance,  and  the  resulting  value 
will  be  quite  near  enough  to  the  actual  measured  or  real 
attenuation  constant.  We  thus  avoid  the  troublesome  calcula- 
tions involved  in  using  the  Campbell  formula. 

As  an  example  of  this  calculation  we  may  take  the  loaded  Anglo- 
French  telephone  cable  laid  in  1910  by  the  General  Post  Office, 
which  is  furthermore  described  in  the  next  chapter  of  this  book. 
The  constants  of  this  cable  as  given  by  Major  O'Meara  are  as 
follows : 

CONSTANTS  OF  THE  UNLOADED  CABLE. 

R  — 14-42          ohms  per  knot  or  nautical  mile  of  loop. 

L=   0-002        henrys 

C=  0-138       microfarad          ,,  „  ,,  „ 

K=   2-4  xlO5  mhos 

n=  750  #  =  27TW  =  4710. 

1  See  Dr.  A.  E.  Kcnnelly,  "  The  Distribution  of  Pressure  and  Current  over 
Alternating  Current  Circuits,"  Harvard  Engineering  Journtd,  1905 — 1906. 


252   PEOPAGATION  OF  ELECTEIC  CUKBENTS 

The  cable  was  loaded  with  coils  having  an  effective  resistance  of 
6  ohms  at  750  p.p.s.  and  an  inductance  of  100  millihenrys.  These 
coils  were  placed  1  knot  (naut.  mile)  apart.  Hence  the  constants 
of  the  loaded  cable  were 

R  =  20*45  ohms  per  knot  loop  of  cable. 

L=   0-1  henry  „  ,,  „ 

C=  0-138          microfarad  ,,  ,,  ,, 

S=   2-4x10"    mhos  ,,  „  „ 

Hence  for  n  —  750  and  p  =  4710  we  have 

*  =  ^418  +  221841.     Also 


Vs*+p*C*  =  10"6  A/576  +  422500. 

/-|  OU 

Again  we  have          VLC  =    /        ,    Zp  =  471, 


Accordingly  the  wave  length  constant 

/-I  OQ 

=  4710^7      {>  =  0-542, 


and  the  wave  length  A  =  27T//3  =  11*6  knots. 

Therefore  the  coils  are  placed  about  11  or  12  to  the  wave  and 
fulfil  the  necessary  condition. 

Then,  since  R  may  be  neglected  in  comparison  with  Lp  and 
S  in  comparison  with  Cp,  we  have 


2 
The  measured  value  was  found  to  be  0'0166. 

6.  Tables  and  Data  for  assisting  Cable  Calcu- 
lations-— The  calculations  necessary  in  connection  with  the 
subject  here  explained  are  facilitated  by  the  possession  of  good 
mathematical  tables  of  various  kinds.  The  reader  will  have 
seen  that  part  of  the  trouble  connected  with  them  depends  upon 
the  necessity  for  constantly  converting  the  complex  expression 
for  a  vector  from  one  form,  a  +  jb,  into  another  form, 
>V/o2  +  62  /  tan"1  I/a,  and  the  reverse.  To  add  or  subtract  two 
complexes  they  must  be  thrown  into  the  form  a  +-  jb,  c  +-  jd, 
and  their  sum  and  difference  are  then  (a  +-  c)  +  j  (b  +  d)  and 


COMPARISON   OF   THEORY  WITH   EXPERIMENT     253 

(a  —  c)  +  j  (b  —  d).    On  the  other  hand,  to  multiply,  divide,  or 
power  them  they  must  be  put  into  the  form  A  j  0,  B  /  0,  where 

A  =  Va?  +  62  and  tan  0  =  b/a,  and  B  =  A/c2  +  d2tan  <£  =  d/c  ; 

j 
and   then   their  product  or  quotient   is  AB  /  0  -\-  $,     ,  /  0—(f>, 


and  square  root  ^4  /  0/2,  etc.  This  process  is  somewhat 
assisted  by  possession  of  good  tables  of  squares  and  square 
roots  of  numbers,  or  by  the  use  of  a  good  slide  rule  or  of  tables 
of  four-figure  logarithms. 

We  can  then  find  from  a  and  b  pretty  quickly  Va?  +  b2.  It 
may  also  be  done  graphically,  but  with  less  accuracy,  by  drawing 
a  right-angled  triangle  whose  sides  are  a  and  6,  and  the  hypo- 
thenuse  is  then  Va?  -j-  b2. 

Very  useful  tables  of  squares  and  square  roots,  as  well  as  of 
circular  and  hyperbolic  functions,  have  been  drawn  up  by  Mr. 
F.  Castle,  and  are  published  by  Macmillan  &  Co.,  St.  Martin's 
Street,  London,  W.C.,  entitled  "  Five-Figure  Logarithmic  and 
other  Tables."  What  is  really  required  is  an  extensive  table 
of  the  logarithms  to  the  base  10  of  hyperbolic  functions,  viz., 
logio  Sinh  u,  logio  Cosh  u,  Log10  Tanhu.from  u  =  0  to  u  —  12, 
and  similar  tables  of  logio  Sin  0,  Logio  Cos  0,  for  various  values 
of  0  in  radians  from  6  =  0  to  0  =  12. 

We  then  require  tables  of  natural  sines,  cosines,  and  tangents. 
If  the  vector  is  given  in  the  form  a  +  jb,  to  convert  to  A  /  0  we 
have  to  find  the  angle  6  whose  tangent  is  b/a,  and  if  given  in  the 
form  A  I  6  we  have  to  find  A  Cos  6  +  jA  Sin  6  to  convert  it  to 

the  other  form. 

Lastly,  we  have  to  provide  tables  of  hyperbolic  functions 
Sink,  Cosh,  Tank,  Seek,  Cosech,  and  Coth.  A  table  of  these 
functions  is  given  in  the  Appendix. 

The  most  troublesome  matter  is  the  calculation  of  the  hyper- 
bolic function  of  complex  angles,  that  is,  finding  the  value  of 
Cosh  (a  -\-jb),  Sinh  (a  +  jb),  etc.  No  tables  of  these  of  any 
great  range  have  yet  been  published.  The  author  understands 
that  such  tables  are  in  course  of  preparation  by  Dr.  A.  E. 
Kennelly,  and  will  be  extremely  valuable.  We  require  to  be  able 
to  find  these  hyperbolic  functions  for  any  vector,  so  that  we  can 


254   PROPAGATION  OF  ELECTRIC  CURRENTS 

enter  the  table  with  values  of  a  and  b  and  find  at  once 
Sinh  (a  +jb),  Cosh  (a  +jb),  etc. 

At  present  the  worker  has  to  calculate  each  case  for  himself 
by  the  formula  given  in  Chapter  L,  viz., 

Sinh  (a  +  jb)  =  Sinh  a  Cos  b  +  j  Cosh  a  Sin  b,  etc.,  etc. 

This  is  a  tedious  business,  but  at  present  there  is  little  available 
assistance. 

The  labour  can  be  somewhat  relieved  by  the  use  of  a  mechanical 
calculator  for  multiplying  and  dividing  numbers.  This  performs 
the  brain-wasting  labour,  and  the  operator  has  then  only  to  put 
the  decimal  point  rightly. 

To  some  small  extent  the  calculations  are  relieved  by  the 
use  of  the  tables  of  Sinh  (a  -\-jb),  etc.,  given  in  Chapter  I. 

The  following  data  for  various  types  of  line  and  receiving 
instruments  will  be  found  very  useful  in  practical  calculations 
and  proposed  undertakings.  They  have  mostly  been  obtained  by 
experience  and  measurements  made  in  the  Investigation  Labora- 
tory of  the  National  Telephone  Company,  and  for  permission  to 
make  use  of  them  here  the  author  is  indebted  to  the  courtesy  of 
Mr.  F.  Gill,  the  Engineer-in-Chief  of  the  National  Telephone 
Company. 

In  all  the  following  tables  the  standard  frequency  n  adopted 
is  796  so  that  2vm  =  5,000.  This  is  sufficiently  near  to 
the  average  telephonic  frequency  to  give  results  useful  in 
practice. 

It  was  agreed  at  the  Second  International  Conference  of 
Engineers  of  Telephone  and  Telegraph  Administrations,  held  in 
Paris,  September,  1910,  that  this  angular  velocity,  p—  5,000, 
should  be  the  standard  one  for  telephonic  measurements,  and 
that  these  should  be  made  with  a  pure  sine  wave  curve  of 
electromotive  force. 

In  the  following  tables  the  abbreviations  used  are  :— 

L.B.  for  local  battery.  An  L.B.  instrument  is  one  supplied 
with  current  from  cells  fitted  locally. 

C.B.  means  central  battery.  By  a  C.B.  termination  is  untfer- 
stood  the  combination  of  a  central  battery  telephone  instrument 
together  with  exchange  cord  circuit  apparatus  which  constitutes 
the  termination  of  the  junction  or  trunk  line. 


COMPARISON   OF   THEORY   WITH   EXPERIMENT     255 

The  following  symbols  are  used  in  the  tables  :— 

E  =  resistance  of  line  per  mile  or  per  kilometre  in  ohms, 

L  =  inductance  of  line  per  mile  or  per  kilometre  in  henrys, 

C  =  capacity  of  line  per  mile  or  per  kilometre  in  farads, 

S  =  dielectric  conductivity  per  mile  or  per  kilometre  in  mhos 
or  reciprocal  ohms, 

p  =  propagation  constant  =  a+j0=  Vli  +  jpL  VS  +  jpC, 

a   -—  attenuation  constant, 

p  =  wave  length  constant, 

A   =  wave  length  =  27T//2, 

W—  wave  velocity  =  p//3, 

ZQ  =  line  impedance  or  initial  sending  end  impedance  = 
VR+jpL/VS+jpC, 

Zr  =  impedance  of  terminal  instrument, 

jfr  =  transmission  equivalent  =  ratio  of  attenuation  constant 
of  the  standard  line  to  attenuation  constant  of  the  line  compared. 
It  gives  the  length  of  the  line  telephonically  equivalent  to  one 
mile  of  the  standard  cable. 

The  quantities  P,  Z0,  Zr,  Zr/Z0,  are  vector  quantities.  Hence 
they  are  expressed  by  stating  their  magnitude  or  size  and  phase 
angle. 

The  following  are  useful  figures  for  terminal  impedances  Zr  of 
National  Telephone  Company's  instruments  : 

L.B.,  II.M.T.  instrument  (S.L.  13),  1060  /60°  ohms. 

No.  1  C.B.  termination,  consisting  of  No.  25  repeater,  super- 
visory relay,  local  line,  and  subscriber's  instrument  with  zero 
local  line,  418  /44°  ohms. 

Ditto  with  300-ohm  line,  730  /30°  ohms. 

The  following  tables  contain  useful  data  and  constants  for 
various  lines  and  cables  : 


256        PROPAGATION   OF   ELECTKIC   CURRENTS 


TABLE  I. — DATA  OF  THE  MORE  IMPORTANT 

British 


Type. 

|  Conductor 
?  Diameter. 

Primary  Constants. 

Propagation 
Constant 

P. 

K 

ohms. 

0 

farad-5. 

L 
heniys. 

S 
mhos. 

OPEN  WIRES  : 

40  Ibs.  per  mile  bronze  . 

70    „          „             „        . 
100    „          „         copper. 

150    „          „            „        . 
200    „          „            „        . 
300    „          „            „        . 
400    „          „             „        . 
600    „          „            „        . 
800    „          „            „        . 

LEAD-COVERED  DRY 
CORE  CABLES  : 

Standard  cable 

Low    capacity     cable, 
Spec'n  No.  127— 
20  Ibs.  per  mile    . 

Cable  to  Spec'n  No.  132— 
6£  Ibs.  per  mile    . 

Cables      to      Spec'n 
No.  125— 
10  Ibs.  per  mile    . 

20    „         „            .        . 
40    „                      .        . 
70    „                      .         . 
100    ,.                     . 
150    „         „            .         . 
200    „         „            . 

1-27 
1-68 
2-01 
2-46 

2-85 
3-48 
4-01 

4-83 

•901 
•508 

•635 
•901 
1-27 
1-68 
2-01 
2-46 

90 
52 
18 
11-9 
9-0 
5-86 
4-50 
2-97 
2-25 

88 

88 
272 

176 

88 
44 
26 
18 
12 
9 

•00750X10-6 
•00786X10-6 
•OOSlOxlO-6 
•00840x10-6 
•00862x10-6 
•00893xlO-6 
•00920x10-6 
•00959  x  10  -6 
•00987x10-6 

•054     x  10-6 

•054     xlO-6 
•0639   xlO  -e 

•0714   xlO-6 

11                  11 
11                  11 
11                  11 
11                  5) 
11                  11 
11                  11 

4-20X10-3 
4-00x10-3 
3-90xlO-s 
3-76x10-3 
3-66x10-3 
3-55x10-3 
3-44xlO-3 
3-31x10-3 
3-22xlO-3 

10x10-3 

1-0x10-3 
negligible 

1-OxlO-3 

11       11 

11             11 
11              11 
11              11 

11         11 

10-6 

11 
» 

5) 
11 
15 
II 

5x10-6 

5  xlO-6 
j»       ?) 

>?       » 
?)       i) 

??          :? 

)>         ?» 
')         D 
11         )» 

•0590  /50°  48' 

•0468  /54°  48' 
•0328  /67°  54' 

•0306/73°  10' 
•0297/76°  15' 

•0289/80°  13' 
•0286/82°    3' 
•0284/84°  19' 

•0283/85°  27' 

•154   /46°   6' 

•154    /46°    6' 

•295   /44°33' 

•251    /45°24' 

•177    /46°13' 

•126    /47°51' 

•0972/50°   3' 
•0816/52°  21' 

•068)  /55°  54' 

•0606/59°    7' 

COMPARISON   OF   THEORY  WITH  EXPERIMENT     257 


TYPES  OF  LINE  FOR  TRANSMISSION  CALCULATIONS. 
Unlit. 


Secondary 

Constants. 

Wive 

Length 
A 
miles. 

Wave 
Velocity 
W    ' 
milrs  per 
second. 

Line 
Impedance 
Zo 
ohms. 

Ratio  *T 
Zo 

I'.li.  Termination. 

L.B. 
Instrument. 

Attenu- 
ation 

Wave 

l.t'imtll 

8- 

Zero  Iccal. 

300W  local. 

•(  I373 
•0270 
•0123 
•00885 
•00706 
•00491 
-00396 
•00281 
•00224 

•137 

•107 
•210 

•176 

•122 
•us  ID 
•0624 
O499 

•0382 
•0311 

•0457 
•0382 
•0304 
•02!>2 
•0238 
•0284 
•0284 
•0282 
•0282 

•111 

•111 
•207 

•179 
•128 
•01)33 
•0745 
•0645 
•0564 
•0520 

137 
HU 
207 
215 
218 
221 
221 
222 
222 

566 

56-6 

30-3 

35-0 
49-0 
67-2 

84-4 
97-6 
112-0 
121-0 

110,000 
131,000 
165  000 
171,000 
174,000 
176,000 
176,000 
177,000 
177,000 

44:900 

44,900 

24.200 

27,900 
39,100 
53,800 
67,100 
77,500 
88,700 
96,200 

l,570\H7b~oT' 

0-266/81°  54' 

0-463  /67C  54' 
0-6  12/63°  43' 

0-674  /  97°  54' 

l,190\33°4a' 

0-351  /77°  43' 

0-890  /  93°  43' 

809  \20°  40' 

0-5  17/66°  40' 

0-902  /50°  40' 

1-31    /  80°  40' 

728  \  1  5°  27' 
688\12°26 

0-5  75/59°  2  7' 

1-00    /45°27' 

1-46    /  75°  27' 

0-609  /56°  26' 
0-648  /52°  28' 

1-06    /42°26' 

1-54    /  72°  26' 

646  \  8°  28' 
622  \  6°  42' 

1-13    /38°28' 
1-17    /36°42' 

1-64    /  68°  28' 

0-672  /50°  42' 
0-704  /4h°  30' 

1-71    /  66°  42' 

594  \  4°  30' 

1-23   /34°30' 

1-79    /'  64°  30' 

575  \  3°  24' 

0-728/47°  24' 

1-27    /33°24' 

1-85    /  63°  24' 

0  733  /86°  50' 

128   /72°50' 

186   /102°50 

1-86    /102°50' 
1-12    /1  04°  33' 

1-51    /103°47' 
2-14    /102°59' 
3-01    /  101°  21' 

571  \42U  50 
571\42°50' 

0-733  /86°  50' 

1-28    /72°50' 

0-452  /88°  33' 

0-790  /74°  33' 
1-04   /73°47' 

924  \44°  33' 

702\43°47' 
497  \42°  59' 

0-596  /87°  47' 

0-84  1/86°  50' 
1-19    /85°21' 

1-47    /72°59' 
2-07    /71°21' 

352\41°21' 

273\39°    9' 

1-53   /83°    9' 

2-C>7    /69°    9' 

3-89    /  99°    9' 

229  \36°  50' 

1-84    /80°50' 

3-18   /6ti°50' 
3-82    /63°17' 

4-63   /    96°  50' 

191\33°17' 

2-19    /77°17' 

5-55    /    93°  17' 

170\30°    5' 

2-46   /74°    5' 

4-29    /60°    5' 

6-24    /  90°    5' 

E.C. 


258       PROPAGATION   OF   ELECTRIC   CURRENTS 


TABLE  II. — DATA  OF  THE  MORE  IMPORTANT 

Metric 


Type. 

Conductor  Weight 
per  kilometre 
(kilograms). 

Primary  Constants. 

Propagation 
Constant 

r. 

R 
ohms. 

C 
farads. 

I 

hemys. 

s 

mhos. 

OPEN  WIRES: 

40  Ibs.  per  mile  bronze  . 

11-3 

56-0 

0-00465  X10-G 

2-61x10  -8 

621x10-6 

•0366  /50°  48' 

70    .. 

19-7 

32-0 

0-00488x10-° 

2-48xlO-3 

11        11 

•0291  /54°  48' 

100    ..          ,.        copper  . 

28-2 

10-9 

0-00503xlO-6 

2-42xlO-3 

11        11 

•0204/67°  54"' 

150    ., 

42-3 

7-30 

0-00522  x  10-6 

2-34x10-3 

11        11 

•0190/73°  10' 

200    „ 

56-4 

5-50 

0-00535xlO-6 

2-28x10-3 

11        11 

•0184/76°  15' 

300    „ 

84-5 

3-64 

0-00554x10-° 

2-20x10-3 

•01  79/80°  IB' 

400    ..          „ 

113 

2-79 

0-00571xlO-6 

2-14x10-3 

11        11 

•0178/82°    3' 

600    ., 

169 

1-82 

0-00595  x  10  -6 

2-06x10-3 

11        11 

•01  76/84°  19' 

800    „ 

226 

1-40 

0-0061  3xlO-6 

2-00x10-3 

11        11 

•01  76/85°  27' 

LEAD-COVERED  DRY 

CORE  CABLES  : 

Standard  cable 

564 

550 

00335xlO-6 

•621  x  10-  3 

31x10-6 

0956/46°   6' 

Low  capacity   cable, 

Spec'n  No.  127— 

20  Ibs.  per  mile 

5-64 

55-0 

0-0335xlO-c 

•621x10-3 

3-1x10-6 

•0956/46°    6' 

Cable  to  Spec'n  No.  132— 

6J  Ibs.  per  mile 

1-83 

169 

0-0396xlO-6 

negligible 

,.       ,, 

•183   /44°33' 

Cables    to     Spec'n 

No.  125— 

10  Ibs.  per  mile 

2-82 

109 

0-0440x10-6 

•621xlO-3 

11       11 

•156    /45°24' 

20    „                       .         . 

5-64 

55-0 

„ 

11         11 

11       i' 

•110   /46°  13' 

40    „                        .         . 

11-3 

27-0 

„ 

11        11 

11       11 

•0781  /47°  51' 

70   „         „,...'. 

19-7 

15-6 

11         11 

11       11 

•0604/50°    3' 

100    .,                        .         . 

28-2 

109 

M            ;i 

11        11 

11          n 

•0507/52°  21' 

150    „                        . 

42-3 

7-30 

11                  •» 

11         11 

11          11 

•0423  /55°  54' 

200    „                       . 

56-4 

5-50 

11            11 

11         11 

11          11 

•0376/59°    7' 

COMPABISON   OF   THEORY   WITH   EXPERIMENT     259 


TYPES  OF  LINE  FOR  TRANSMISSION  CALCULATIONS. 


Secondary 
Constants. 

Wave 
Length 
A. 
kilo- 
metres. 

Wave 
Velocity 
W 
kilometres 
per 
second. 

177,000 
210,000 
265,000 
276,000 
280,000 
283,000 
283,000 
285,000 
285,000 

72.300 

72,300 
39,000 

45,000 
63,000 
86,700 
108,000 
125,  000 
143,000 
155,000 

Line 
Impedance 
Zo 
ohms. 

Ratio  ZJL. 
Zo 

C.B.  Termination. 

L.B. 
Instrument. 

Attenu- 
ation 
a. 

\V  ave 
Length 
0- 

Zero  local. 

300W  local. 

•0232 

•0168 
•00764 
•00549 
•00438 
•00304 
•00246 
•01117.") 
•00139 

0663 

•or,*;:} 

•131 

•109 
•0768 

•0524 
•0388 
•0810 
•0237 
•0193 

•02S4 
•0238 
•0189 
•01J-2 
•0179 
•0176 
•0176 
•017:, 
•0175 

0639 

•0689 

•128 

•112 
•071)4 
•().•)  7!) 
•Q462 
•0401 
•0351 
•<i:;23 

222 
264 

334 
348 
351 
366 
356 
359 
369 

911 

91-1 

is-s 

66-4 

78-9 
108 
136 
157 
180 
195 

0-266  /81°  54' 

0-4  63/67°  54' 

O-f.74/  97°  54' 

1,570\37°54' 

1,190  \33°  43' 
809  \20°  40' 
728\15°27' 

0-351  /77°  43' 

0-61  2/63°  43' 

0-890  /  93°  43' 

0-51  7/66°  40' 

0-902  /50°  40' 
1-00    /45°  27' 

1-31    /  80°  40' 

0-575  /59°  27' 

1-46    /  75°  27' 
1-54    /  72°  26 

688  \1  2°  26' 

0-609  /56°  26' 

1-06    /42°26' 

64  6  \  8°  28' 

0-648  /52°  28' 

1-13    /38°28' 

1-61    /  68°  28' 

622  \  6°  42' 

0-672  /50°  42' 

1-17   /36°42' 

1-76    /  66°  42' 

594  \  4°  30' 

0-704/4  8°  30' 

1-23    /34°30' 

1-79    /  64°  30' 

575  \  3°  24' 
571  \42°  50' 

0-728/47°  24' 

1-27    /33°24' 

1-85    /  63°  24' 

0  733  /86°  50' 

1-28    /72°50' 

186   /  102°  50 

0-733  /86°  50' 

1-28    /72°50' 
0-790/74°  33' 

1-86    /1  02°  50' 
1-12    /  104°  33' 

1-51    /1  03°  4  7' 

571  \42°50' 
924\44°33' 

702\43°47' 

0-452  /88°  33' 

0-596/87°  47' 

1-04   /73°47' 

497  \  42°  59' 

0-841  /86°  59' 

1-47    /72°59' 

2-14    /1  02°  59' 

352\41°21' 

1-19   /85°21' 
1-53    /83°    9' 

2-07   /71°2l' 

3-01    /101°21' 

273\39°    9' 
229\36°50' 
191\33°17' 

2-67    /H9°    9' 

3-89    /  99°    9' 

1-84    /80°50' 

3-18    /6<i°50' 

4-63    /  96°  50' 

2-19    /77°17' 

3-82   /63°17' 

5-55    /  93°  17' 

170\30°    5' 

2-46    /74°    5' 

4-29   /60°    5' 

6-24    /  90°    5' 

s  2 


260        PROPAGATION   OF   ELECTRIC   CURRENTS 


TABLE  III. --DATA  OF  THE  LESS  IMPOIITANT  TYPES  OF  LINE  FOE  TRANSMISSION 

CALCULATIONS. 

British     Units. 


Type. 

Primary  Constants. 

Secondary 
Constant. 

B 

ohms. 

C 

farads. 

L 

henrys. 

8 

mhos. 

Attenuation 
a. 

LEAD  -  COVERED      DRY     CORE 

CABLES  : 

Cables  to  Spec'n  No.  126  — 

20  Ibs.  per  mile    .... 

88 

•0822xlO-6 

1-0  x  10-8 

5xlO-6 

•131 

40    ..     "                  .... 

44 

5                    5 

5            55 

•0905 

70   „                       .... 

26 

n 

5             55 

•0669 

100   ,,                      .... 

18 

1                    1 

55 

•0534 

150   „                       .... 

12 

5            55 

•0409 

200   „                       .... 

9 

,                    , 

JJ 

5             55 

•0333 

Cable  to  Spec'n  No.  10  — 

1  2£  Ibs.  per  mile 

144 

0-54  xlO~6 

55               55 

55            55 

•138 

KUBBER-COVERED        DRY       CORE 

AERIAL  CABLES  : 

Spec'n  No.  134— 

6£  Ibs.  per  mile 

272 

•0785xlO-6 

negligible 

55            55 

•232 

Special,    weight    under    1     Ib. 

per  foot  — 

6J  Ibs.  per  mile 

272 

•0987xlO-c 

i; 

55            55 

•260 

Spec'n  No.  130— 

10  Ibs.  per  mile 

176 

•0775xlO-6 

1-0  xlO-3 

55            55 

•183 

Spec'n  No.  20A— 

12^  Ibs.  per  mile     . 

144 

•0700x10-6 

55               55 

55             55 

•157 

Spec'n  Nos.  20  and  131  — 

20  Ibs.  per  mile       .         .        . 

88 

•0700x10-° 

55               55 

55            55 

•122 

MISCELLANEOUS     WIRES     AND 

CABLES  : 

1 

22/15  V.I.R.  opening-out  . 

146 

•250x10-6 

l-3x!0-3 

infinity 

•297 

20/12  twin  V.IJL       . 

87 

•225x10-° 

55               55 

M 

•213 

20/10    V.IJL  cable,  with   steel 

suspender        .... 

87 

•300xlO-e 

55               55 

?? 

•246 

20/10  twin  V.  I.E.  leading-in  and 

opening-out     .... 

87 

•200  xlO-6 

55               55 

55 

•201 

Silk  and  cotton  cable  — 

9£  Ibs.  per  mile 

192 

•lOOxlO-6          negligible 

55 

•219 

COMPARISON   OF   THEORY  WITH   EXPERIMENT     261 


TABLE  IV. — DATA  OF  THE  LESS  IMPORTANT  TYPES  OF  LINE  FOR  TRANSMISSION 

CALCULATIONS. 

Metric  I'n'tt*. 


Type. 

Conductor 
Weight  PIT 
kilometre 
(kilo- 
grams). 

Primary  Constants. 

Secondary 
Constant. 

E 

ohms. 

c 

farads. 

L 
lu'iirys. 

S 
mhos. 

Attenuation 
ft. 

LKAD-COVERED  DRY  CORE 

CABLES  : 

Cables  to  Sper'n  No.  126  — 

20  Ibs.  per  mile     . 

5-61 

55-0 

0'05  lOx  10  -fi 

•621  xlO-3 

3-1  X  10-6 

•OS  14 

•Id    .             ,             . 

11-3 

270 

?i 

•0562 

7(>    . 

11I-7 

15-6 

55                ?) 

•0415 

IdO    , 

28-2 

10-i) 

•0332 

150    , 

423 

730 

??                11 

•0254 

200   ,             ,             .         . 

50-4 

5--0 

11                11 

•0207 

Cable  to  Spec'n  No.  10— 

12^  Ibs.  per  mile     . 

3-52 

89-0 

0-054  x  10  -6 

11                11 

„          „ 

•0857  . 

RUBBER  -COVERED     DRY 

CORE  AERIAL  CABLES  : 

Spec'n  No.  134  — 

6£  Ibs.  per  mile 

1-83 

169 

0-0187xlO~G 

negligible 

•  i         11 

•144 

Special,  weight  under  1  Ib. 

per  foot  — 

6£  Ibs.  per  mile 

1-83 

169 

0-0613  x  I0-(i 

,, 

11         11 

•161 

Spec'n  No.  130— 

10  Ibs  per  mile 

2-82 

109 

0-0  181  x  10-6 

•621  x  10-3 

•i         11 

•114 

Spec'  11  No.  20A— 

12£  Ibs.  per  mile     . 

352 

89'0 

00435X  !'J-fl 

)!                11 

11          11 

•0975 

Spec'n  Nos.  20  and  131  — 

20  Ibs.  per  mile      . 

5-64 

:,:>•<> 

0-0435  x  10-6 

11 

11          11 

•0758 

MISCELLANEOUS    WIRES 

AND  CABLES  : 

1 

22/15  V.J.Jf.  opening-out. 
20/12  twin  V.I.R.      . 

3-40 
5-70 

91-0 
54-0 

0  155x  10-6 
0-140x  10-6 

•808xlO-3 

infinity 

11 

•184 
•132 

2(>/lo    V.I.  It.    cable,  with 

steel  suspender 

5-70 

54-0 

0-186  xlO*6 

11        11 

,, 

•153 

20/10  twin   ]  .1.11.  leading- 

in  and  opening-out 

5-70 

51-0 

0-124  x  'O-6 

11        11 

>i 

•125 

Silk  and  cotton  cable  — 

It1,  Ibs.  per  mile 

2-60 

119 

0-0620x10-° 

negligible 

11 

•136 

262        PROPAGATION   OF   ELECTEIC   CURRENTS 


TABLE  V. — TRANSMISSION  EQUIVALENTS. 


Trans- 

Reciprocal 

Trans- 

Reciprocal 

Type. 

mission 

of 

Type. 

mission 

of 

Equivalent, 

Equivalent. 

Equivalent. 

Equhalent. 

OPEN  WIRES  : 

LEAD-COVERED  DRY  CORE 

40  Ibs.  per  mile  bronze     . 

2-830 

0-353 

CABLES  (continued)  : 

70    „ 

.V890 

0-257 

Cables  to  Spec'n  No.  126 

100    ,,          ,,       copper     . 

8-440 

0-118 

(continued)  — 

150    „ 

11-680 

0-0853 

150  Ibs.  per  mile 

2-588 

0-386 

200    „ 

14-710 

0-0680 

200     „                        . 

3-168 

0-316 

300    „ 

21-000 

0-0476 

Cable  to  Spec'n  No.  10— 

400  „     ;, 

26-050 

0-0384 

12£  Ibs.  per  mile 

0-775 

1-290 

600    „ 

800    „ 

36-750 
45-750 

0-0272 
0-0218 

RUBBER-COVERED  DRY 
CORE  AERIAL  CABLES  : 

LEAD-COVERED  DRY  CORE 
CABLES  : 
Standard  cable    . 

1000 

1-000 

Spec'n  No.  134— 
•6J  Ibs.  per  mile 
Special,  weight  under  1  lb. 

P                      A 

0-460 

2-173 

Low  capacity  cable. 
Spec'n  No.  127  — 
20  Ibs.  per  mile 
Cable  to  Spec'n  No.  132  — 
6^  Ibs.  per  mile 
Cables  to  Spec'n  No.  125- 
10  Ibs.  per  mile 
20    „                         . 

1-000 
0-509 

0-605 

0-872 

1-000 
1-965 

1-654 
1-147 

per  root  — 
6J  Ibs.  per  mile 
Spec'n  No.  130— 
10  Ibs.  per  mile 
Spec'n  No.  20A  — 
12|  Ibs.  per  mile     . 
Spec'n  Nos.  20  and  131  — 
20  Ibs.  per  mile 

0-410 
0*582 

0-678 
0-880 

2-440 
1-718 
1-475 
1-136 

40    „                          .         . 

1-262 

0-792 

MISCELLANEOUS  WIRES 

70    „                           . 

1-705 

0-587 

AND  CABLES  : 

100    „                         . 

2-130 

0-470 

22/15  V.I.It,  opening  out  . 

0-359 

2-785 

150    „                           . 

2-775 

0-360 

20/12  twin  V.I.R. 

0-497 

2-010 

200    „                           .         . 

3-400 

0-294 

20/10    V.I.R.    cable,    with 

Cables  to  Spec'n  No.  126— 

steel  suspender 

0-430 

2-325 

20  Ibs.  per  mile 

0-810 

1-235 

20/10  twin  V.I.R.  leading- 

40    „                          .         . 

1-175 

0-850 

in  and  opening-out 

0-528 

H92 

70    „                         . 

1-590 

0-629 

Silk    and     cotton     cable. 

100    „                           .         . 

1-990 

0-502 

9J  Ibs.  per  mile 

0-486 

2-058 

CHAPTER  IX 

LOADED  CABLES  IN  PRACTICE 

1.  Modern  Improvements  in  Telephonic  Cables 
and  Lines. — The  result  of  nearly  twenty  years'  investigations 
by  mathematical  physicists  and  practical  telephonists,  starting 
from  the  date  of  Mr.  Oliver  Heaviside's  first  fertile  suggestions, 
has  been  to  effect  a  great  improvement  in  the  transmitting 
powers  of  telephonic  lines  by  working  in  the  direction  indicated 
by  Heaviside,  viz.,  that  an  increase  in  the  inductance  of  the  line 
would  reduce  attenuation  and  distorsion.  Although  many 
schemes  were  put  forward  for  increasing  the  inductance  of  the 
line  by  enclosing  it  in  iron,  and  several  alternative  proposals,  such 
as  those  of  Professor  S.  P.  Thompson,  for  placing  across  it 
inductive  shunts,  it  cannot  be  said  that  the  suggestions  bore 
much  practical  fruit  until  after  Professor  Pupin's  important 
contribution  to  the  subject  by  his  proposal  to  locate  the  induct- 
ance in  equispaced  loading  coils,  coupled  with  a  practical  rule 
for  their  effective  spacing.  The  result  of  this  has  been  that 
practical  experience  has  now  accumulated  to  a  considerable 
extent  in  connection  with  the  two  methods  of  carrying  out  the 
Heaviside-Pupin  recommendations,  viz.,  increasing  the  induct- 
ance of  the  line  by  uniform  loading  and  increasing  it  by  loading 
coils  at  intervals. 

The  uniform  loading  consists  in  wrapping  or  enclosing  the 
copper  conductor  in  iron  wire  in  such  a  manner  that  the 
magnetic  flux  produced  around  it  by  the  telephonic  currents  is 
increased,  with  a  corresponding  increase  in  the  effective  induct- 
ance, and  therefore  diminution  of  the  attenuation  constant,  with 
more  or  less  reduction  in  the  distorsion  of  the  wave  form  produced 
by  the  line. 

Three  cases  present  themselves  for  consideration,  viz.,  aerial 


264        PROPAGATION   OF   ELECTRIC   CURRENTS 

or  overhead  lines,  underground  cables,  and  submarine  telephonic 
cables.  We  shall  describe  briefly  what  has  been  attempted  and 
achieved  in  each  case.  The  improvement  of  telephony  con- 
ducted through  overhead  or  aerial  conductors  has  been  effected 
solely  through  the  use  of  loading  coils.  Aerial  lines  are  not 
adapted  for  uniform  loading.  It  would  involve  a  great  increase 
in  the  weight  per  mile  and  necessitate  stronger  cables  and  more 
expensive  supports,  and  also  offer  greater  surface  to  wind  and  snow. 
The  writer  is  not  aware  that  it  has  ever  been  tried.  On  the 
other  hand,  aerial  lines  are  well  suited  for  loading  coils,  since 
these  can  be  attached  at  intervals  to  the  posts  which  carry  the 
line. 

So  far,  then,  uniform  loading  has  been  restricted  to  under- 
ground cables  and  to  submarine  cables,  whilst  the  non-uniform 
loading  or  application  of  loading  coils  has  been  extensively 
tried  on  underground  lines,  and  in  a  few  cases,  but  with  great 
success,  in  the  case  of  under-water  cables. 

In  respect,  however,  of  the  improvement  gained  or  to  be 
gained  in  the  case  of  aerial  lines  and  underground  or  under- 
water cables  respectively,  the  following  remarks  of  Dr.  Hammond 
V.  Hayes  in  a  paper  read  before  the  St.  Louis  International 
Electrical  Congress  are  important l: 

11  In  the  case  of  cables  there  is  a  distinct  improvement  in  the 
"  quality  of  the  transmission  produced  by  the  introduction  of 
"  the  loading  coils,  the  voice  of  the  speaker  being  received  more 
"  distinctly.  The  high  insulation  which  can  be  maintained  at 
"  all  times  on  cable  circuits  renders  it  possible  to  introduce 
"  loading  coils  upon  the  circuits  without  danger  of  materially 
"  augmenting  leakage  losses.  The  marked  diminution  in 
"  attenuation,  the  improvement  in  quality  of  transmission,  and 
"  the  ease  with  which  inductance  coils  can  be  placed  on  cable 
"  circuits  without  introducing  other  injurious  factors,  such  as 
"  leakage  or  cross-talk  with  other  circuits,  renders  the  use  of 
"  loaded  cable  circuits  especially  attractive." 

"  The  reduction  of  attenuation  that  can  be  obtained  by  the 
"  introduction  of  loading  coils  on  air-line  circuits,  even  under 

1  See  reprint  of  this  paper  in  The  Electrician,  Vol.  LIV.,  p.  362,  December  16th, 
1904,  "Loaded  Telenhone  Lines  in  Practice." 


LOADED   CABLES   IN   PRACTICE  265 

<k  theoretically  perfect  conditions,  is  less  than  can  be  obtained  on 
"  cable  circuits.  This  difference  in  the  effectiveness  of  loading 
"  between  the  two  classes  of  circuits,  as  far  as  attenuation  is  con- 
"  cerned,  can  be  explained  by  the  fact  that  on  a  cable  circuit  the 
"  capacity  is  large  and  the  inductance  of  the  circuit  itself  is 
"  practically  negligible,  due  to  the  proximity  of  the  two  wires  of 
"  the  pair.  On  aerial  circuits,  on  the  other  hand,  the  distance 
"  between  the  outgoing  and  return  wire  is  such  as  to  make  the 
"  capacity  of  the  circuit  much  less,  and  its  inductance  much 
"greater.  This  larger  self-induction  of  the  open-wire  circuit 
"  operates  to  decrease  the  attenuation,  and,  as  it  were,  to  rob  the 
"  loading  coils  of  part  of  their  usefulness.  Again,  the  insulation 
"  of  an  aerial  circuit  cannot  be  maintained  as  high  as  that  of  a 
"  cable  circuit,  so  that  the  added  inductance  due  to  the  intro- 
"  duction  of  loading  coils  upon  the  line  tends  to  increase  the 
"  losses  due  to  leakage." 

"  Moreover,  there  is  not  the  same  improvement  in  the  quality 
"  of  transmission  on  a  loaded  aerial  circuit,  as  compared  with  a 
"  similar  circuit  unloaded,  as  is  found  between  loaded  and 
"  unloaded  cables.  Initially,  open-wire  circuits  are  practically 
"  free  from  distorsion,  whereas  the  distorsion  on  cable  circuits  of 
"  long  length  is  considerable.  The  addition,  therefore,  of  loading 
"  coils  to  aerial  circuits  cannot  be  expected  to  effect  any  improve- 
"  merit  in  the  quality  of  transmission,  whereas  in  the  case  of 
"  cables  the  introduction  of  the  additional  inductance  renders 
"  the  circuits  practically  distortionless  and  effects  a  marked 
"  improvement  in  the  clearness  of  the  transmitted  speech." 

It  is  perhaps  well  to  point  out  here  that  the  two  qualities 
essential  in  telephonically  transmitted  speech  are  sufficient 
fondness  or  volume  of  sound  and  clearness  or  distinctness.  Both 
these  qualities  are  necessary  for  intelligibility.  There  may  be 
clearness,  but  the  speech  may  be  so  faint  that  only  people  with 
exceptionally  good  hearing  can  comprehend  it.  On  the  other 
hand,  there  may  be  loudness  but  not  clearness,  and  the  speech  is 
then  also  not  intelligible.  The  loss  of  volume  is  due  to  the 
attenuation  generally,  but  the  loss  of  distinctness  to  the  differ- 
ence in  the  attenuation  of  the  different  harmonic  frequencies  and 
consequent  distorsion  of  the  wave  form. 


266        PROPAGATION   OF   ELECTEIC   CURRENTS 

In  the  case  of  the  aerial  lines  the  want  of  loudness  in  the 
transmitted  sound  is  chiefly  due  to  the  resistance  of  the  line,  and 
in  so  far  as  this  is  the  cause  it  cannot  be  much  alleviated  by  the 
introduction  of  inductance.  It  is  only  the  attenuation  which 
arises  from  distributed  capacity  which  can  be  reduced  by  added 
inductance.  In  cables,  on  the  other  hand,  the  predominant 
cause  of  the  attenuation  is,  generally  speaking,  capacity,  and  it 
is  therefore  appropriately  remedied  by  the  introduction  of 
inductance. 

Nevertheless  experience  shows  that  some  advantage  is  gained 
by  the  introduction  of  loading  coils  into  aerial  lines. 

2.  The  Introduction  of  Loading  Coils  into 
Overhead  or  Aerial  Lines.— The  effect  of  introducing 
inductance  coils  of  low  resistance  into  aerial  lines  has  now  been 


FIG.  1.— Loading  Coil  used  in  the  Berlin -Magdeburg  Aerial 
Telephone  Line. 

tried  on  several  long  lines,  and  found  to  be  an  advantage.  These 
coils  take  the  form  of  a  closed  iron  circuit-choking  coil  having  a 
laminated  or  iron  wire  core,  covered  over  with  a  low  resistance 
wire. 

The  general  form  of  coil  and  core  and  leading-in  sleeve  may 
be  seen  from  the  diagram  in  Fig.  1,  which  represents  the  coils 


LOADED  CABLES  IN  PEACTICE 


267 


used  on  the  first  German  line  so  treated,  viz.,  the  Berlin- 
Magdeburg  line,  150  km.  in  length.  The  coils  were  mounted  on 
an  arm  together  with  a  vacuum  lightning  arrester,  mounted  in 
parallel  with  the  coil. 

After  a  preliminary  trial  on  the  Berlin-Magdeburg  line  it  was 
decided  to  equip  a  longer  line,  and  the  Berlin-Frankfort-on-Main 
was  chosen,  as  the  distance  is  about  580  km.  (=  360  miles). 
A  new  bronze  wire,  2'5  mm.  in  diameter,  was  accordingly  run. 
Also  between  the  terminal  points  there  existed  two  other  bronze 
wires,  one  4  mm.  in  diameter  and  the  other  5  mm.  All  lines 
were  double  wire  lines.  The  inductance  or  loading  coils  were 
inserted  every  5  km.  on  the  2'5-mm.  line.  The  effective 
resistance  of  each  coil  was  8'7  ohms,  and  its  inductance  O'll 
henry.  Hence  the  coils  add  3*48  ohms  to  the  resistance,  and 
0*044  henry  to  the  inductance  per  kilometre  of  loop  or  distance. 
The  general  result  as  regards  speech  transmission  was  that, 
whereas  before  loading  the  speech  volume  on  the  2'5-mm.  line" 
was  of  course  less  than  that  on  the  5-mm.  and  4-mm.  lines, 
after  loading  the  loaded  2'5-mm.  line  was  better  than  the  4-mm. 
unloaded  line,  but  not  quite  so  good  as  the  5-mm.  unloaded  line. 

The  following  are  the  constants  and  attenuation  constants  of 
these  four  lines  at  a  frequency  of  900  : 


Resistance 

Inductance 

Capacity       Attenuation 

LINE. 

If. 

L 

6'in               Constant 

in  ohms. 

in  henrys. 

microfarads. 

a. 

Bronze   \\ire  5   mm. 

1-92 

0-00186 

0-0063 

0-00176 

diameter  unloaded 

Ditto  4  mm.  diameter 

3-00 

0-00194 

0-0060 

0-00262 

Ditto  2-5  mm.  diameter 

7-70 

0-00214 

0-0055 

0-00591 

unloaded 

2-5     mm.      diameter 

11-18         0-0461 

0-0055 

0-00193 

loaded  every  5  km. 

/»'  is  the  effective  resistance  in  ohms  per  kilometre  of  loop ; 
L  is  the  inductance  in  henrys  per  kilometre  of  loop ;  C  is  the 


268        PEOPAGATION   OF   ELECTRIC   CURRENTS 

capacity  wire  to  wire  in  microfarads  per  kilometre  of  loop ;  a  is 
the  attenuation  constant  per  kilometre  of  loop. 

The  loaded  2'5-mm.  line  is  equivalent  to  an  unloaded  4'7-mm. 
line  of  the  same  material. 

The  product  of  the  attenuation  constant  and  the  length  of  the 
line,  called  the  attenuation  length,  is  as  follows  : 

1.  For  the     5-mrn.  line  al  =  0'95, 

2.  For  the     4-mm.  line  al  =  1*52, 

3.  For  the  2'5-mrn.  line  unloaded     al  =  3*43, 

4.  For  the  2'5-mm.  line  loaded         al  =  1*12. 

The  smaller  the  attenuation  length  al  the  better  the  speech- 
transmitting  qualities  of  the  line.  It  is  generally  considered 
that  a  line  permits  excellent  talking  when  al  is  not  more  than  2*5, 
and  fair  speech  when  al  does  not  exceed  3*5.  Hence  the  2*5-mm. 
unloaded  line  is  efficient,  but  becomes  better  on  loading. 

It  has  been  agreed  that  with  an  ordinary  copper  line  joined 
directly  to  the  telephonic  apparatus  the  relation  between  speech 
and  attenuation  length  al  is  as  follows : 

Speech  up  to  Attenuation  lengths  <«/. 

equal  to 

Very  good  ,,     ,,  2*5 

Good  „     „  3-5 

Practical  limit  at  4-8 

This  corresponds  to  about  forty-six  miles  of  the  National 
Telephone  Company's  standard  cable  when  using  the  standard 
type  of  central  battery  instrument  and  circuit  at  either  end  of 
the  line,  and  a  subscribers'  line  of  300-obms  resistance. 

The  result  therefore  of  loading,  in  the  above  manner,  the 
Berlin-Frankfort-on-Main  2'5-mm.  line  has  been  to  effect  a 
sensible  increase  in  the  speech  efficiency  of  the  line. 

Previously  to  the  equipment  of  the  above  long  distance  line 
experiments  had  been  tried  on  the  Berlin-Magdeburg  overhead 
line,  2-mm.  bronze  wire,  150  km.  in  length. 

This  line  was  equipped  with  loading  coils  having  an  effective 
resistance  of  six  ohms  and  an  inductance  of  0'08  henry  placed 
every  4  km.  The  result  was  better  speech  than  that  over 
a  3-mm.  bronze  wire  180  km.  in  length  running  between  the 
same  places. 


LOADED  CABLES  IN  PKACTICE 


269 


Also  between  Berlin  and  Potsdam  (32*5  km.)  on  certain  lines 
coils  of  4*1  ohms  and  0*062  henry  were  introduced  every  1/3  km. 
The  result  was  an  increase  in  the  inductance  per  km.  of  two 
hundredfold  and  a  reduction  of  the  attenuation  constant  to 
one-sixth  of  that  of  the  unloaded  line. 

In  loading  an  aerial  line  or  a  cable  it  is,  however,  necessary  to 
make  arrangements  to  avoid  losses  by  reflection  at  the  point 
where  the  loaded  line  joins  on  to  an  unloaded  or  terminal  line. 
It  has  already  been  ex- 
plained that  when  a  i-o 
telephone  wave  passes 
across,  the  junction  of 
two  lines  which  differ 
considerably  in  induct- 
ance or  capacity  per 
unit  of  length  there  is 
a  reflection  of  energy 
which  acts  to  produce 
an  increased  attenuation 
in  certain  cases.  In 
practice  the  effect  of 
reflection  is  very  con- 
siderable, particularly 
when  the  loaded  section 
is  relatively  not  long. 
Theoretically  this  reflec- 
tion can  be  eliminated 
by  the  introduction  of 
a  perfect  transformer  at 


0-4 


0-3 


C-2 


c-i 


400      COO      800     1,000 
Length  of  Line. 


1,400 


FIG.  2. — Curves  showing  effect  of  loading 
Coils  on  an  aerial  line  435  Ibs.  to  the 
mile  (H.  V.  Hayes). 


every  point  of  discontinuity  in  the  line ;  practically  it  is  best  over- 
come by  the  employment  of  what  is  called  a  terminal  taper.  This 
consists  in  a  series  of  several  inductance  coils  placed  near  the  ends 
of  the  loaded  section,  each  one  having  somewhat  less  inductance 
than  the  preceding  one  and  less  than  that  of  the  coils  in  the  main 
loaded  section.  Hence  the  inductance  per  mile  or  per  kilometre 
is  not  suddenly  changed,  but  reduced  gradually  or  tapered  off 
from  that  in  the  loaded  section  to  that  in  an  unloaded  line.  The 
spacing  of  the  coils  in  the  taper  is  the  same  as  that  in  the 


270        PBOPAGATION   OF   ELECTEIC   CURRENTS 


main   part   of   the   loaded  line.      This   taper   is   introduced   at 
both  ends. 

The  effect  of  taper  and  loading  is  well  shown  in  some  curves 
which  have  been  given  by  Dr.  Hammond  V.  Hayes  in  an 
interesting  paper1  entitled  "  Loaded  Telephone  Lines  in  Practice." 
The  coils  used  were  toroidal  in  shape,  about  10  inches  in  diameter 
and  4  inches  high,  and  had  an  effective  resistance  of  15*5  ohms 

at  2,000  periods  per 
second,  but  only  2'4 
ohms  steady  resistance 
and  an  inductance  of 
0'25  henry.  On  aerial 
circuits  such  coils  are 
placed  about  two  miles 
or  so  apart,  so  as  to  give 
an  inductance  of  about 
O'l  henry  per  mile. 

The  curves  in  Fig.  2 
show  the  effect  of  such 
loading  on  an  aerial  line 
weighing  435  Ibs.  to  the 
mile.  Curve  1  shows 
the  decrease  in  current 
at  the  receiving  end  for 
various  lengths  when  the 
line  is  unloaded,  curve  2 
when  the  transmitting 
and  receiving  instru- 
ments are  connected  to  the  loaded  line  without  taper,  and 
curve  3  the  same  when  the  line  is  tapered  at  both  ends. 

The  curves  in  Fig.  3  show  the  same  results,  but  for  a  line 
consisting  of  wire  176  Ibs.  per  mile,  and,  as  before,  curve  1  shows 
the  attenuation  of  the  unloaded  line,  curve  2  of  the  loaded 
untapered  line,  and  curve  3  the  loaded  and  tapered  line.  These 
curves  show  clearly  that  for  short  lengths  of  line  loading  is  not 

1  Read  before  Section  6  of  the  St.  Louis  International  Electrical  Congress,  1004  ; 
also  see  T/te  Electrician,  December  16th,  1904,  Vol.  LIV.,  p.  362,  or  Se-lenre  Abstract*, 
VII.  B,  Abs.  2,968,  1904, 


400 


600      800    l.OCO 
Lcnyth  of  Line. 


1,200    1,400 


FIG.  3. — Curves  showing  effect  of  loading 
on  an  aerial  line  176  Ibs.  to  the  mile 
(H.  Y.  Hayes). 


LOADED  CABLES  IN  PRACTICE 


271 


beneficial,  but,  on  the  contrary,  reduces  the  received  current  con- 
siderably. This  is  because  the  added  resistance  increases  the 
attenuation  constant  at  first  more  than  the  added  inductance 
reduces  it. 


3.    Loaded      Underground     Cables.— As     already 
remarked,  the  benefits  to  be  expected  from  loading  a  line  either 
continuously  or  at  intervals  are  likely  to  be  more  pronounced  in 
the  case  of  cables  than 
of  aerial  lines,   for   the 
reason  that  the  capacity 
per     mile     is     always 
greater   in   the   case   of 
cables,  and  therefore  its 
peculiar    effect   in    pro- 
ducing  attenuation  and 
distorsion  is  capable  of 
remedy     by     suitably 
introduced  inductance. 

Moreover,  in  under- 
ground cables  there  are 
no  particular  difficulties 
involved  in  introducing 
the  inductance  coils 
when  spaced  impedance 
is  added.  The  coils  can 
be  of  any  convenient 
size  and  can  be  located 
in  small  watertight  chambers  placed  at  regular  intervals  on 
the  line. 

Dr.  Hammond  V.  Hayes  has  given  in  the  same  paper  (loc.  cit.) 
some  curves  for  loaded  cables  similar  to  those  above  given  for 
aerial  lines. 

Fig.  4  shows  the  result  of  loading  a  telephone  cable  having  a 
pair  of  wires  each  0'03589-inch  diameter  and  a  resistance  of 
96  ohms  per  mile  of  circuit  (double  wire  circuit).  The  capacity 
is  0'068  microfarad  per  mile.  The  inductance  added  by  the 
loading  coils  amounted  to  about  0'6  henry  per  mile. 


40        60        80        100      120 
Length  of  Line. 

FIG.  4. — Curves  showing  effect  of  loading 
on  a  Telephone  Cable  (H.  V.  Hayes). 


272        PEOPAGATION   OF   ELECTEIC  CUKEENTS 


Curve  1  in  Fig.  4  shows  the  attenuation  on  the  unloaded 
cable,  curve  2  the  same  for  the  loaded  cable  without  taper,  and 
curve  3  the  attenuation  for  the  loaded  and  tapered  line.  It  will 
be  seen  that  the  effect  of  loading  without  taper  is  to  reduce 
greatly  the  sending  end  current  and  to  increase  the  received 
current  beyond  a  certain  length  of  line. 

The  effect  of  loading  with  taper  is  to  reduce  somewhat  the 

sending  end  current,  but 
to  greatly  increase  the 
received  current  beyond 
short  distances  when 
compared  with  the  un- 
loaded line. 

A  comparison  of 
curves  2  and  3  shows 
how  great  a  factor  the 
reflection  losses  are  be- 
tween the  terminal 
apparatus  and  the  loaded 
line  and  how  important 
it  is  to  employ  taper 
to  reduce  these  losses. 
In  Fig.  5  are  given  two 
curves.  Curve  1  is  the 
attenuation  curve  of 
unloaded  line,  and 


10 


0-9 


0-8 


07 


06 


0-5 


0-4 


0-3 


0-2 


01 


60        80       100 
Length  of  Line. 


140 


FIG.  5. — Curves  showing  the  effect  of  loading 
on  a  Telephone  Cable  (H.  V.  Hayes). 


an 

curve  2  for  the  same 
line  lightly  loaded  and  without  taper.  It  is  seen  that  the  reflec- 
tion losses  are  much  reduced,  and  that  when  no  taper  is  employed 
it  is  easily  possible  to  overload  the  line  detrimentally. 

The  results  of  loading  as  far  as  the  cable  itself  is  concerned 
can  be  predicted  by  means  of  the  formulae  given,  but  it  is  less 
easy  to  foresee  the  exact  results  when  tapering  is  not  employed. 
Hence  in  those  numerous  cases  in  which  a  loaded  trunk  cable 
has  aerial  lines  connected  on  at  both  ends  the  importance  of 
introducing  suitable  taper  is  very  great. 

The  necessity  for  maintaining  good  insulation  on  loaded 
cables  is  discussed  in  a  later  section  of  this  chapter.  Meanwhile 


LOADED  CABLES   IN   PRACTICE 


273 


it  may  be  stated  that  loaded  underground  cables  have  been 
extensively  employed  by  the  National  Telephone  Company  in 
Great  Britain  with  great  advantage. 

The  type  of  impedance  coil  adopted  after  careful  experiment  is 


FIG.  6. — Loading  or  Inductance  Coil 
(without  case)  as  used  by  the 
National  Telephone  Company  of 
Great  Britain. 

shown  in  Fig.  6.  It  consists  of  a  choking  coil  having  a  closed 
magnetic  circuit  formed  of  fine  soft  iron  wire  and  overlaid  with 
silk-covered  insulated  copper  wire.  The  finished  toroidal  coil 
has  an  overall  diameter  of  about  4*5  to  5  inches,  and  a  central 
aperture  of  about  T5  inches,  and  a  depth  of  nearly  2  inches. 


FIG.  7. — A  Diagram  showing  the  mode  of  Winding  the 
Loading  Coil  in  two  parts  and  their  insertion  in  the 
two  sides  of  the  Cable. 

The  effective  resistance  of  such  a  coil  may  be  from  3*5  to  15 
ohms  for  currents  of  1,000  frequency,  and  the  inductance  may 
be  from  0*06  to  0'25  henry.  Each  coil  is  wound  in  two  parts, 
one-half  being  inserted  in  the  lead  and  one  in  the  return  (see 
Fig.  7). 

B.C.  T 


274        PEOPAGATION   OF   ELECTKIC   CUERENTS 

The   following  table  gives   the    data   of    some   of    the    coils 

employed : 

DATA  FOE  LOADING  COILS. 


Loading. 

Spacing 
Interval  in 
miles 
between  coils. 

Steady 
Resistance 
in  ohms. 

Effective 

in  ohms  at 
a  frequency 
of  1,000. 

Inductance 
in  henrys. 

Very  light  . 

5-75 

1-18 

3-5 

•059 

Light  .... 

2-5 

2-84 

7-5 

•133 

Medium 

1-75 

3-97 

11-7 

•176 

Heavy 

1-25 

6-11 

15-7 

•252 

The  toroidal  coils  are  enclosed  in  a  watertight   iron  case,  and 


FIG.  8.  — A  Brick  Pit  for  containing  an  Iron 
Case  in  which  are  a  number  of  Loading 
Coils,  as  constructed  by  the  National 
Telephone  Company. 

a  number  of  these  coils  may  be  placed  in  a  brick  pit  and  inserted 
in  the  circuit  of  cables  passing  through  the  pit  (see  Fig.  8). 

4.  Loaded  Submarine  on  linden-water  Tele- 
phone Cables.— Whilst  there  is  little  or  no  difficulty  in 
introducing  loading  coils  into  aerial  lines  or  underground  cables, 


LOADED  CABLES  IN  PRACTICE       275 

the  problem  of  applying  these  methods  to  under-water  cables 
presents  peculiar  difficulties.  Any  considerable  enlargements  on 
a  submarine  cable  must  not  only  add  to  its  weight  and  to  the 
strains  experienced  during  laying,  but  may  also  increase  the 
difficulties  of  laying  very  greatly.  It  was  therefore  with  some 
hesitation  that  telegraphic  engineers  approached  this  particular 
work,  and  it  was  only  when  the  great  and  certain  improvements 
made  by  loading  land  lines  had  clearly  established  beyond  doubt 
that  submarine  telephony  must  be  equally  improved,  if  the 
mechanical  difficulties  of  making  and  laying  such  a  cable 
could  be  overcome,  that  the  matter  was  taken  seriously  in  hand. 
Even  then  it  was  felt  that  the  difficulties  of  manufacture 
and  laying  of  a  continuously  loaded  cable  might  be  less  than 
those  of  a  loaded  cable,  and  the  first  efforts  seem  to  have  been  in 
this  direction. 

The  continuously  loaded  cable  has,  however,  two  disadvan- 
tages as  compared  with  the  non-uniformly  loaded  cable.  It  is 
undoubtedly  more  expensive  to  make,  and  it  is  not  possible  to 
predict  with  any  certainty  the  attenuation  constant  of  a  cable  so 
made.  This  arises  from  the  impossibility  of  determining 
beforehand  the  permeability  of  the  iron  wire  which  is  laid  over 
the  core  to  increase  its  permeability,  and  also  from  changes 
in  that  permeability  during  and  after  laying,  and  also  from  the 
unknown  increase  in  the  effective  resistance  of  the  core  which 
results  from  the  iron  wire  envelope  due  to  hysteresis  and  eddy 
currents. 

The  general  construction  of  a  continuously  loaded  cable  is  as 
follows  :  The  copper  core  is  insulated  and  overlaid  with  several 
windings  of  fine  iron  wire,  and  this  is  insulated  either  with 
gutta-percha  or  with  paper.  If  the  latter  is  used,  then  a 
continuous  lead  covering  has  to  be  put  over  the  paper  to  keep 
it  dry,  and  over  that  protecting  layers  of  jute  or  hemp  and  then 
the  usual  steel  armouring.  The  iron  wire  laid  over  the  copper 
then  increases  the  inductance  to  a  certain  extent  not  easy  to 
foretell  accurately.  Cables  on  this  plan  have  been  laid  in 
Germany  and  Holland,  and  the  following  details  and  table  are 
taken  from  a  valuable  paper  by  Major  O'Meara,  C.M.G., 
Engineer-in-chief  of  the  British  Postal  Telegraphs,  read  before 

T  2 


276        PKOPAGATION   OF   ELECTKIC  CUEKENTS 

the  Institution  of  Electrical  Engineers  of  London  in  November, 
1910,  "  On  Submarine  Cables  for  Long-distance  Telephone 
Circuits." 

Major  O'Meara  states  that  the  first  continuously  loaded  cable 
having  the  copper  conductor  wrapped  with  a  layer  of  0'008-inch 
iron  wire  on  the  plan  devised  by  Mr.  C.  E.  Krarup,  the 
Engineer-in-chief  of  the  Danish  Telegraph  Service,  appears  to 
have  been  that  laid  by  the  Danish  Government,  in  November, 
1902,  between  Elsinore  and  Helsingborg.1  Mechanical  and 
electrical  data  of  this  cable  are  given  in  the  table.  The  dielectric 
was  gutta-percha,  and,  except  in  respect  of  the  iron  wrapping, 
the  cable  did  not  differ  materially  from  the  ordinary  type  of 
submarine  cable.  This  was  followed,  as  will  be  seen  from  the 
table,  by  various  paper-insulated  cables  having  the  conductors 
wrapped  with  a  single  layer  of  0*012-inch  iron  wire.  The  cable 
distinguished  by  the  letter  E  in  the  table  on  p.  278,  was 
laid  in  July,  1904.  Each  copper  conductor  consists  of  a 
central  wire  about  0*089  inch  in  diameter  surrounded  by  three 
copper  strips  each  0*094  inch  wide  and  0*020  inch  thick.  The 
sectional  area  of  the  copper  is  approximately  0*0124  square 
inch,  and  the  weight  per  knot  285  Ibs.  The  iron  wrapping  con- 
sists of  three  layers  of  0'008-inch  wire,  and  the  insulator  is 
gutta-percha  having  an  external  diameter  of  0*354  inch.  The 
four  cores  are  laid  up  with  an  inner  serving  of  tanned  jute 
and  an  outer  serving  of  tarred  jute  yarn  to  a  diameter  of  1*18  inch, 
and  sheathed  with  fifteen  galvanised  iron  wires  of  roughly  trape- 
zoidal section.  The  external  covering  appears  to  be  the  usual 
tarred  yarn  and  compound. 

The  electrical  constants  of  the  cable  per  knot  from  Mr.  Krarup's 
figures  are  given  on  p.  277. 

Of  the  paper-insulated  lead-covered  cables  the  Dano-German 
telephone  cable  laid  between  Fehmarn  and  Lolland  in  1907  may 
be  taken  as  representative.  The  copper  conductor  with  its 
triple  soft  iron  wire  wrapping  is  precisely  similar  to  that  used  in 
the  Seeland-Samso-Jutland  cable  described  above.  The  insulator 
consists  of  paper  cord  laid  on  in  an  open  spiral  followed  by  a 

1  "  Modernc  Telefonkabler,"  by  C.  E.  Krarup,  Elektroteltnilieren,  December 
10th,  1904. 


LOADED  CABLES  IN  PEACTICE 


277 


close  wrapping  of  paper  ribbon  up  to  a  diameter  of  0*303  inch. 
Four  of  the  cores  so  formed  are  stranded  together  with  the 
necessary  worming  and  then  covered  with  paper  to  a  diameter  of 
0'787  inch.  The  diagonal  distance  apart  of  the  cores,  centre  to 
centre,  is  0*413  inch.  The  core  after  being  thoroughly  dried  is 
next  sheathed  with  two  layers  of  lead  alloyed  with  3  per  cent, 
of  tin,  each  layer  being  0'055  inch  thick.  The  lead  sheath  is 
seamless,  watertight,  and  continuous  throughout  the  entire 
length  of  the  core.  Outside  the  lead  sheath  is  a  double  layer  of 
asphalted  paper  and  a  layer  of  jute  and  compound.  The  armour 
consists  of  thirteen  galvanised  iron  wires  or  strips  of  trapezoidal 

,.       /0-315+0-252  .     ,\ 

section  f—    — 5 —      -  x  0-157  square  inch),  and  over  this  is  a 

double  layer  of  jute  and  compound. 


Resistance. 

Ohms  per  Knot  of 
Conductor. 


Capacity. 

Microfarads  per  Knot  of 
Conductor. 


Inductance. 


Steady 
Current. 

Alternating 
Current, 
«  =  900. 

Steady 
Current. 

Alternating 
Current. 

With  Iron. 

Without 
Iron. 

3-971 

4-175 

0-4983 

0-4454 

8-07 

0-93 

To  prevent  the  destruction  of  the  cable  by  the  puncture  of  the 
lead  sheath  at  any  point,  solid  plugs  1  metre  (3*28  feet)  long  are 
inserted  at  every  150  metres  (164  yards).  The  constants  of  the 
cable  are  as  follows  : 

Eesistance  per  knot  of  loop,  8-924  ohms         ...)  Continuous 
Capacity  per  knot  of  loop,  0-0872  microfarad...  J     current. 

Capacity  per  knot  of  loop,  0'0770  microfarad... 

(    current. 

Inductance  per  knot  of  loop,  18-26  to  18-09  millihenrys. 

The  table  on  p.  278,  taken  by  permission  from  Major  O'Meara's 
paper  (loc.  cit.\  gives  the  details  of  some  continuously  loaded 
cables. 


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LOADED   CABLES   IN   PEACTICE  279 

A  great  difference  seems  to  exist  between  the  attenuation 
constants  of  continuously  loaded  cables  as  actually  measured 
and  that  which  theory  would  predict  so  far  as  the  measured  data 
allow.  Thus  for  a  cable  made  for  the  Danish  Telegraph  Service 
continuously  loaded  with  three  layers  of  iron  wire  0'0079-inch 
diameter  the  observed  attenuation  constant  was  0*0296,  whereas 
that  calculated  from  certain  data  as  to  the  permeability  of  the 
iron  was  0!0197. 

An  additional  objection  to  continuous  loading  by  envelopes  of 
iron  wire  is  that  it  increases  the  capacity  by  increasing  the 
diameter  of  the  conductor.  Also,  in  the  opinion  of  experts,  its  cost 
is  about  twice  as  great  as  that  of  Pupin  loading  for  equal  effect. 

Accordingly  attention  has  more  recently  been  directed  to  the 
question  of  designing  under-water  cables  loaded  with  Pupin  coils 
at  intervals,  and  two  successful  examples  of  this  are  the  loaded 
lead-covered  telephone  cable  laid  by  Messrs.  Siemens  and  Halske 
across  Lake  Constance  in  1906  and  the  loaded  submarine  tele- 
phone cable  laid  by  the  British  Postal  Telegraph  Department  in 
1910  across  the  English  Channel  between  Abbotscliff,  in  England, 
and  Grisnez,  in  France. 

The  Lake  Constance  cable  is  about  9J  miles  long.  The 
maximum  depth  of  the  cable  is  138  fathoms,  at  which  depth  the 
pressure  is  about  25  atmospheres.  The  cable  contains  seven 
speaking  circuits,  and  these  cores  are  enclosed  in  insulation  and 
covered  with  steel  armour,  over  which  a  continuous  coating  of 
lead-tin  alloy  is  pressed  and  then  a  jute  covering  and  a  second 
armour.  The  loading  coils  are  cylindrical  and  are  slipped  over 
the  cable  and  connected  in  circuit  with  one  conductor.  The 
capacity  wire  to  wire  is  0*038  microfarad  per  kilometre,  and 
the  inductance,  including  loading  coils,  is  0*20  henry,  and  the 
effective  resistance  is  33*5  ohms  per  kilometre  at  a  frequency 
of  900.  The  attenuation  constant  is  0*0072  per  kilometre.  For 
details  of  the  work  of  laying  and  other  information,  which,  how- 
ever, is  rather  sparse,  the  reader  is  referred  to  an  article  on  this 
cable  in  The  Electrician,  Vol.  LIX.,  p.  217,  1907. 

The  Anglo-French  loaded  Four-core  telephone  cable  of  1910 
laid  by  the  British  Post  Office  across  the  English  Channel, 
represents  at  present  (in  1911)  the  latest  achievement  in  the 


280   PROPAGATION  OF  ELECTRIC  CURRENTS 

laying  of  loaded  submarine  cables,  and  the  following  is  a 
description  of  this  cable  taken  verbatim  from  Major  O'Meara's 
valuable  paper  on  the  subject  : 

"  The  features  of  the  device  for  loading  in  the  accepted  tender 
are  as  follows  : 

"  The  two  double  coils  required  for  the  four  conductors  of  the 
cable,  each  coil  being  of  slightly  less  than  6  ohms  resistance  and 
having  an  inductance  of  010  henry  at  750  periods  per  second, 
are  inserted  at  intervals  of  1  knot  (1*153  miles),  but  the  two 
coils  nearest  the  ends  of  the  cable  are  inserted  at  a  distance  of 
only  half  a  knot  from  the  terminal  apparatus,  as  experiments 
have  shown  that  in  this  arrangement  reflection  losses  are  con- 
siderably reduced.  Each  double  coil  consists  of  two  windings  on 
the  same  iron  core,  and  one  winding  is  connected  in  series  with 
each  conductor  (see  Fig.  9).  By  this  means  the  gradual  change 
in  permeability  in  the  iron  core  due  to  ageing  will  not  affect  the 
balance  in  the  two  limbs  of  the  telephone  circuit.  Each  coil  is 
protected  with  a  sheet  of  metal  foil  in  order  to  exclude  all  possi- 
bility of  the  silk  covering  of  the  wires  of  the  coils  absorbing 
moisture  from  the  cylindrical  envelope  of  gutta-percha  in  which 
they  are  contained.  The  cores  of  the  cable  are  connected  to  the 
envelope  at  its  two  ends  by  tapered  solid  gutta-percha  joints. 
The  diameter  at  the  centre  of  the  envelope  is  3  inches,  and  at  the 
cores  where  the  joints  terminate  1  inch.  An  annular  rubber 
distance-piece  is  inserted  between  the  two  coils  of  a  set  to  give 
greater  flexibility.  The  total  length  of  the  joint  is  30'75  inch. 
As  the  diameter  of  the  cable  at  the  points  where  the  coils 
are  inserted  is  increased,  a  larger  number  of  sheathing  wires 
are  required  at  those  points  than  over  the  conductors  alone. 
This  difficulty  is  ingeniously  overcome  by  starting  a  second 
layer  of  sheathing  wires  over  the  cores,  about  27  feet  from 
the  centre  of  the  coil  envelope,  and  gradually  working  them  into 
a  single  layer  with  those  over  the  bulge.  Finally,  they  are 
terminated  as  a  second  layer  again  over  the  cores  at  a  distance 
of  about  27  feet  from  the  centre  of  the  coil  envelope.  The  method 
adopted  in  inserting  the  coils  (British  Patent  Specification 
No.  5,547,  March,  1907)  will  perhaps  be  understood  from  the 
diagrams  (Fig.  9)." 


LOADED  CABLES  IN  PRACTICE 


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282   PROPAGATION  OF  ELECTRIC  CURRENTS 

The  following  is  the  Post  Office  specification  for  the  cable, 
arrived  at  after  most  careful  consideration  of  the  problem  by  the 
technical  experts  of  the  department : 

SPECIFICATION  FOR  ANGLO-FRENCH  SUBMARINE  TELEPHONE 

CABLE. 

1.  Conductors. — The   conductor   of   each  coil  shall  be  of   an 
approved   stranded  type,  shall  weigh  not  less  than  160  Ibs.  per 
knot,  and  shall  at  a  temperature  of  75°  F.  have  a  resistance  not 
higher  than  7 '452  standard  ohms  per  knot  for  a  conductor  of 
this  gauge.     The  lay  of  the  stranded  conductor  shall  be  left- 
handed. 

2.  Insulator  or  Dielectric. — The  conductor  of  each  coil  shall  be 
insulated  by  being  covered  with  three  alternate  layers  of  Chatter- 
ton's  compound  and  gutta-percha,  beginning  with  a  layer  of  the 
said  compound,  and  no  more  compound  shall  be  used  than  may 
be  necessary  to  secure  adhesion  between  the  conductor  and  the 
layers  of  gutta-percha.     The  dielectric  on  the  conductor  of  each 
coil  shall  weigh  not  less  than  300  Ibs.  per  knot,  making  the  total 
weight  of   the  conductor  of  each  coil  when  covered  with   the 
dielectric  not  less  than  460  Ibs.  per  knot. 

3.  Inductive  Capacity. — The  inductive  capacity  of  each  coil  of 
such  insulated  conductor  (hereinafter  called  the  core)  shall  not 
exceed  0*275  microfarad  per  knot,  and  this  shall  apply  equally  to 
the  completed  cable. 

4.  Insertion   of  Loading    Coils. — The    loading    coils    will    be 
inserted  so  that  diagonal  cores  in  the  cable  will  be  used  to  form 
a  loop  or  pair,  each  pair  of  cores  to  be  fitted  with  loading  coils 
equally  spaced  at  such  distances  apart  and  of  such  inductance 
and  effective  resistance  as  will  make 

(a)  The  volume  of  speech  transmitted  over  a  pair  of  wires  in 
the  completed  and  laid  cable  at  least  equal  to  that 
through  one-seventh  of  the  same  length  of  standard 
cable,  not  including  terminal  losses 1  ; 

1  Standard  cable  is  that  having  a  wire-to-wire  capacity  for  each  pair  of  wires  of 
0-054  microfarad  per  statute  mile,  a  loop  resistance  of  88  ohms  per  statute  mile, 
and  an  average  insulation  resistance  of  not  less  than  200  megohms  per  statute  mile 
wire  to  wire. 


LOADED   CABLES   IN   PEACTICE  283 

(/>)  The  quality  of  speech  or  articulation  not  inferior  to  that 
of  the  speech  throughout  the  standard  cable  equivalent1 
of  the  loaded  cable  pair. 

5.  Interference. — The  two  loaded  cable  pairs  to  be  free  from 
telephonic   induction  or  interference,   the  one  from  the  other, 
and  also  from  external  disturbance  from  a  contiguous  cable. 

6.  Labelling. — Each  coil  of  core  before  being  placed  in  the 
temperature  tank  for  testing  shall  be  carefully  labelled  with  the 
exact  length  of  conductor  and  the  exact  weight  of  copper  and 
dielectric  respectively  which  it  contains. 

7.  Insulation  Iicsistance. — The   insulation  resistance   of   each 
coil  of  core,  after  such  coil  shall  have  been  kept  in  water  main- 
tained at  a  temperature  of  75°  F.  for  not  less  than  twenty-four 
consecutive  hours  immediately  preceding  the  test,  shall  be  not 
less  than   400   nor  more  than  2,000  megohms  per  knot  when 
tested  at  that  actual  temperature,  and  after  electrification  during 
one  minute.    The  electrification  between  the  first  and  the  second 
minutes  to  be  not  less  than  3  nor  more  than  8  per  cent.,  and  to 
progress  steadily.     The  insulation  to  be  taken   not   less   than 
fourteen  days  after  manufacture. 

Each  coil  of  core  may  be  subjected,  before  the  ordinary 
insulation  test  is  taken,  to  an  alternating  electromotive  force  of 
5,000  volts  and  100  complete  periods  per  second  for  fifteen 
minutes. 

8.  Preservation. — The  core  shall  during  the  process  of  manu- 
facture be    carefully  protected   from  sun    and    heat,  and  shall 
not  be  allowed  to  remain  out  of  water. 

9.  Joints. — All  joints  shall  be  made  by  experienced  workmen, 
and  the  contractor  shall  give  timely  notice  to  the  Engineer-in- 
chief   or   other   authorised    officer   of   the   Postmaster-General 
whenever  a  joint  is  about  to  be  made,  in  order  that  he  may  test 
the  same.     The  contractor  shall  allow  time  for  a  thorough  testing 
of  each  and  every  joint  in  the  insulated  trough  by  accumulation, 
and  the  leakage  from  any  joint  during  one  minute  shall  be  not 
more  than  double  that  from  an  equal  length  of  the  perfect  core. 

1  By  the  standard  cable  equivalent  of  any  loop  is  meant  the  number  of  statute 
miles  of  loop  in  a  standard  cable  through  which  the  same  volume  of  speech  is 
obtained  as  through  the  loop  under  test. 


284        PBOPAGATION   OF   ELECTRIC   CURRENTS 

10.  Taping  and  Serving. — The  cores  to  be  four  in  number,  and 
to  be  stranded  with  a  left-handed  lay,  and  during  the  process  of 
stranding  be  wormed  with  best  wet  fully  tanned  jute  yarn,  so 
that  the  whole  may  be  as  nearly  as  possible  of  a  cylindrical  form, 
and  shall  then  be  covered  (1)  with  cut  cotton  tape  prepared  with 
ozokerit   compound,  (2)    with  pliable  brass  tape  0*004  inch  in 
thickness  and  1  inch  in  width,  and  (3)  with  another  serving  of 
cotton  tape,  similar  to  the  first,  the  lap  in  each  case  being  not 
less  than  0'250  inch. 

The  cores,  prepared  as  above  specified,  shall  then  be  served 
with  best  wet  fully  tanned  jute  yarn,  sufficient  to  receive  the 
sheathing,  hereafter  specified,  and  no  loose  threads  shall,  in  the 
process  of  sheathing,  be  run  through  the  closing  machine.  The 
cores  so  served  shall  be  kept  in  tanned  water  at  ordinary  tempera- 
ture, and  shall  not  be  allowed  to  remain  out  of  water  except  so 
far  as  may  be  necessary  to  feed  the  closing  machine. 

11.  Sheathing. — The  served  core  to  be  sheathed  with  sixteen 
galvanised  iron  wires,  each  wire  having  a  diameter  of  280  mils, 
or  within  3  per   cent,   thereof  above  or  below  the  same.     The 
breaking  weight  of  each  wire  to  be  not  less  than  3,500  Ibs.,  with 
a  minimum  of  ten  twists  in  6  inches.     The  length  of  lay  to  be 
18  inches,  and  to  be  left-handed. 

The  wire  to  be  of  homogeneous  iron,  well  and  smoothly 
galvanised  with  zinc  spelter.  The  galvanising  will  be  tested  by 
taking  samples  from  any  coil  or  coils,  and  plunging  them  into  a 
saturated  solution  of  sulphate  of  copper  at  60°  F.,  and  allowing 
them  to  remain  in  the  solution  for  one  minute,  when  they  will  be 
withdrawn  and  wiped  clean.  The  galvanising  shall  admit  of  this 
process  being  four  times  performed  with  each  sample  without 
there  being,  as  there  would  be  if  the  coating  of  zinc  were  too  thin, 
any  sign  of  a  reddish  deposit  of  metallic  copper  on  the  wire.  If, 
after  the  examination  of  any  particular  quantity  of  iron  wire,  10 
per  cent,  of  such  wire  does  not  meet  all  or  any  of  the  foregoing 
requirements,  the  whole  of  such  quantity  shall  be  rejected,  and 
no  such  quantity  or  any  part  thereof  shall  on  any  account  be 
presented  for  examination  and  testing,  and  this  stipulation  shall 
be  deemed  to  be  and  shall  be  treated  as  an  essential  condition  of 
the  contract.  Before  being  used  for  the  sheathing  of  the  cable, 


LOADED   CABLES   IN   PEACTICE  285 

the  wire  shall  be  heated  in  a  kiln  or  oven,  just  sufficiently  to 
drive  off  all  moisture,  and  whilst  warm  shall  be  dipped  into  pure 
hot  gas-tar  (freed  from  naphtha).  The  iron  wire  so  dipped  shall 
not  be  used  for  sheathing  the  cable  until  the  coating  of  gas-tar  is 
thoroughly  set.  No  weld  or  braze  in  any  one  wire  of  the  sheatli 
shall  be  within  six  feet  of  a  weld  or  braze  in  any  other  wire.  All 
welds  or  brazes  made  during  the  manufacture  of  the  cable  shall 
be  regal vanised  and  retarred. 

12.  Compound  and   Serving. — The   sheathed    cores    shall    be 
covered   with    two  coatings  of   compound  and  two  servings  of 
three-ply  jute  yarn,  the  said  compound  being  placed  between  the 
two  servings  and  over  the  outer  serving  of  yarn  aforesaid,  the  two 
servings  of  yarn  to  be  laid  on  in  directions  contrary  to  each 
other. 

The  compound  referred  to  in  this  paragraph  shall  consist  of 
pitch  85  per  cent.,  bitumen  12  J  per  cent.,  and  resin  oil  2^  per 
cent.,  and  the  yarn  referred  to  shall  be  spun  from  the  best 
quality  of  jute,  and  shall  be  saturated  with  gas-tar  freed  from 
acid  and  ammonia,  the  yarn  being  thoroughly  dried  after 
saturation  and  before  being  used,  so  as  to  have  no  superfluous 
tar  adhering. 

13.  Measurement   and   Marks. — A    correct    indicator  shall  be 
attached  to    the  closing  machine,  and  a  mark  to  be  approved 
by  the  Engineer-in-chief  shall    be  made    on  the    cable  at  the 
termination  of  each  knot  of  completed  cable,  and  also  over  each 
joint  or  set  of  joints. 

14.  Laying. — If  the  tender  for  laying  be  accepted,  the  contrac- 
tors shall   provide   the    necessary   cable-laying    ship    and    all 
appliances  and  all  apparatus  in   connection    therewith   for  the 
laying  and  testing   of  the  cable   during  the  laying  operations. 
Facilities  must  be  provided  for  inspection  of  the  work,  if  con- 
sidered   necessary,    by    an    officer   of    the    Postmaster- General 
during  the  progress  of  the  laying  operations. 

The  cable  to  be  laid  over  the  course  shown  by  the  dotted 
red  line  on  the  accompanying  Admiralty  chart,  or  as  hereafter 
agreed  upon. 

On  completion  of  the  laying  operations  the  spare  cable  left 
on  board  is  to  be  delivered  at  the  Post  Office  Cable  Depot, 


286        PEOPAGATION   OF   ELECTEIC   CUERENTS 

Dover,  or  paid  out  and  buoyed  in  the  sea  near  Dover,  as  may 
be  directed   by  the  Engineer-in-chief. 

15.  The  contractors  are  required  to  guarantee  that  the  com- 
pleted cable  shall  reach  and  maintain  the  standard  laid  down 
in  the  specification,  and  before  final  acceptance  the  cable  shall 
be  subject  to  such  tests  and  experiments  as  the  Postmaster- 
General  may  deem  necessary  during  the  manufacture,  laying, 
and  for  a  period  of  thirty  consecutive  days  from  the  completion 
of  the  latter. 

Major  O'Meara  states  (loc.  cit.)  that  "  the  investigations 
that  had  been  made  left  little  doubt  concerning  the  balance 
of  advantages  in  favour  of  the  '  coil '  loaded  type  of  cable 
from  the  electrical  standpoint,  but  as  the  expenditure  involved 
was  very  great,  and  as  it  was  felt  that  the  main  difficulty  in 
connection  with  this  type  of  cable  would  be  in  safely  Ia37ing 
the  cable  at  the  bottom  of  the  sea,  it  was  considered  that 
special  precautions  were  necessary  to  ensure  that  the  responsi- 
bility for  any  defects  that  might  be  disclosed  after  it  had  been 
laid  should  be  definitely  traced  to  the  responsible  party.  To 
afford  the  necessary  protection  to  the  department,  it  seemed 
desirable  to  stipulate  in  the  specification  that  the  manufacturers 
of  the  cable  should  also  undertake  to  lay  it,  and  to  hand  it 
over  in  situ.  This  course  was  approved  by  the  Postmaster- 
General,  and  the  invitations  to  tender  were  issued  on  these 
lines.  The  conditions  were  accepted  by  Messrs.  Siemens  Bros. 
&  Co.,  who  were  the  successful  tenderers. 

"  It  will  be  recognised  that  the  mechanical  problem  in  connec- 
tion with  this  type  of  cable  was  more  difficult  to  solve  than  the 
electrical  problem,  as  it  was  necessary  that  the  part  of  the  cable 
containing  the  coils  should  be  so  designed  that  it  could  be  paid 
over  the  sheaves  of  the  cable-ship  without  any  risk  of  damage  to 
the  coils  themselves.  However,  Major  O'Meara  said  he  was  glad 
to  say  that  the  manufacturers  succeeded  in  solving  this  problem 
in  a  most  satisfactory  manner. 

"  The  cable  was  under  the  constant  supervision  of  the  Post  Office 
Engineering  Department  during  the  period  of  its  manufacture, 
and  electrical  tests  were  carried  out  from  time  to  time.  On 
January  18th,  1910,  after  the  completion  of  the  cable,  measure- 


LOADED  CABLES  IN  PEACTICE       287 

merits  to  determine  its  attenuation  constant  were  made  at  the 
works  of  Messrs.  Siemens  Bros.  &  Co.  at  Woolwich.  The  con- 
ductors of  the  cable  were  joined  up  so  as  to  provide  a  metallic 
circuit  of  41*704  knots,  and  in  order  to  get  rid  of  terminal  effects 
artificial  cable  was  joined  to  tha  ends  of  the  loaded  cable 
as  shown  in  Fig.  10. 

Current  was  supplied  to  this  circuit  by  a  generator  giving  1*585 
volts  at  a  frequency  of  750  alternations  per  second.  Eeadings 
were  taken  on  a  thermo-galvanometer  placed  successively  at  A 
and  13,  and  the  attenuation  constant  was  calculated  by  the 
formula  /a  =  Ii  f~  al- 

"With  ten  miles  of  '  standard '  cable  (attenuation  constant 
0'1187  per  knot)  at  each  end  of  the  circuit  the  current  values  at 


Artificial 
Cable 


A<- 41-704    Knots 

FIG.  10. 

.1  were  found  to  be  0*327  milliampere,  and  at  B  0*172  milliam- 
pere, a  therefore  being  0'0154. 

"With  fifteen  miles  of  '  standard '  cable  at  each  end  of  the 
circuit  the  current  values  at  A  were  found  to  be  0*212  milliam- 
pere, at  B  0*110  milliampere,  from  which  we  similarly  obtain 
a  —  0-0152. 

"  The  volume  of  the  speech  transmitted  over  the  loaded  cable 
was  also  compared  with  that  over  an  artificial  "  standard  "  cable, 
the  electrical  constants  of  which  are  known.  The  result  of  these 
tests  indicated  that  the  attenuation  constant  of  the  loaded  cable 
was  0*0147." 

The  table  on  p.  288,  given  by  Major  O'Meara,  supplies  the 
details  of  the  primary  constants  of  this  cable  both  with  loading 
coils  inserted  and  without  them,  and  it  also  shows  the  attenuation 
constants  before  and  after  loading. 

Mr.  W.  Dieselhorst  was  entrusted  by  Messrs.  Siemens  Bros, 
with  the  actual  operation  of  laying  the  cable,  and  Mr.  F.  Pollard, 
Submarine  Superintendent,  Dover,  was  detailed  to  watch  the 
interests  of  the  Post  Office. 


288        PROPAGATION   OF   ELECTRIC   CURRENTS 


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LOADED  CABLES  IN  PKACTICE 


289 


• 


- 


B.C. 


290        PEOPAGATION   OF  ELECTKIC   CURKENTS 

For  the  full  details  of  the  laying  of  this  cable  and  the  manner 
in  which  the  engineering  difficulties  were  overcome  in  the  manu- 
facture and  laying  by  the  contractors,  Messrs.  Siemens  Bros., 
the  reader  must  consult  Major  O'Meara's  admirable  paper  on 
the  subject  in  the  Journal  of  the  Institution  of  Electrical 
Engineers. 

The  photograph  reproduced  in  Fig.  11  is  taken  by  permission 
from  Major  O'Meara's  paper  (loc.  cit.),  and  represents  the  passing 
of  a  loading  coil  in  the  1910  Anglo-French  cable  over  the  sheaves 
of  the  cable-ship  Faraday  during  the  process  of  laying  the 
cable.  It  will  be  seen  that  the  type  of  loading  coil  adopted  does 
not  render  the  cable  to  any  extent  cumbersome  and  unhandable. 
The  constants  of  the  cable  and  some  numerical  values  con- 
nected therewith  both  for  the  unloaded  cable  and  for  the  cable 
with  loads  are  very  approximately  as  follows  : 

Unloaded  Cable  Loaded  Cable 

per  nautical  mile.  per  nautical  mile. 

E=14-42  ohms,  ^  =  20'45  ohms, 

L  =  0-002  henry,  £  =  01  henry, 

C=-138  X  10-6  farad,  C=  138  x  1Q-6  farad, 

S=24  X  10-5  mhos.  S=2-4  x  10~5  mhos. 


Hence  for  the  loaded  cable  we  have 

Vti*+p*L*=  ^418  +  2217841 

2C2  =10-6  A/576  +  422,500 


Therefore  for  the  loaded  cable 

a=y  -£-  nearly  =^/_=.016  (approximately); 


=  4,710  =  "542. 


Hence  X=-|Wll'6nauts, 

1=204-5     |=169 

IRC  /s    1Q   ,'BC. 

and  a=V-2-A/C=1Vir 


LOADED   CABLES   IN   PRACTICE 


291 


The  loading  coils,  being  1  naut  apart,  are  therefore  at  the  rate 
of  eleven  or  twelve  per  wave  for  the  standard  wave  length, 
corresponding  to  a  frequency  of  ahout  800,  and  the  spacing 
complies  with  Pupin's  law. 

As  regards  the  practical  improvement  introduced  by  the 
loading  coils  in  the  above  cable  the  following  quotation  from 
Major  O'Meara's  paper  (loc.  cit.)  is  interesting  and  important. 
He  said : 

"  The  cable  has  been  under  continuous  observation  since  it  was 
laid,  and  a  large  number  of  tests  have  been  carried  out.  Par- 
ticulars of  some  of  them  are  given  in  an  appendix.  It  has 
fortunately  been  possible  to  obtain  independent  testimony  on  the 
question  of  the  increase  in  the  range,  and  in  the  improvement  in 
the  quality  of  speech  transmitted  by  means  of  the  loaded  cable 
as  compared  with  a  similar  cable  unloaded.  Speech  tests  were 
made  in  July  last  by  Messrs.  W.  E.  Cooper,  W.  Duddell,  F.E.S., 
W.  Judd,  and  J.  E.  Kingsbury,  and  the  results  are  interesting. 
The  cable  was  looped  at  the  French  end  (Cape  Grisnez),  and  the 
English  ends  were  connected  to  two  telephone  sets,  one  installed 
in  the  cable  hut  at  Abbot's  Cliff  and  the  other  in  the  coastguard 
look-out  shelter  some  100  feet  distant.  Graduated  artificial  cables 
were  provided  so  that  the  listener  at  the  cable-hut  could  insert 
various  values  of  the  '  standard  '  cable  into  the  circuit  until  his 
own  limit  of  satisfactory  audibility  was  reached.  It  was  possible 
to  insert  the  *  standard '  cable  values  equally  at  the  two  ends  of 
the  cable  (i.e.,  so  as  to  form  a  symmetrical  circuit  in  relation  to 
the  submarine  cable),  or  unequally,  as  desired.  The  results 
shown  in  the  table  below  were  obtained. 


Observer  listening. 

Old  Cable. 

New  Cable. 

Gain 

vy 
Jsew 

Cable. 

Added  Length  of  Standard 
Cable. 

Added  Length  of  Standaid 
Cable. 

W.  R.  Cooper  .        .      *  . 
W.  Duddell      . 

W.  Judd  .... 
J.  E.  Kingsbury 

24  miles  symmetrical 
24  miles  symmetrical 

26  miles  symmetrical 
26  miles  symmetrical 

48  miles  symmetrical 
1    40  miles  symmetrical 
•j    50  miles  symmetrical 
(    55  miles  at  one  end 
40  miles  symmetrical 
40  miles  symmetrical 

Miles. 
24 
16 
26 
21 
14 
14 

u  2 


292        PKOPAGATION   OF  ELECTRIC   CURRENTS 

"  The  mean  gain  by  the  use  of  the  new  cable  is  therefore  seven- 
teen miles  of  '  standard '  cable  for  the  standard  of  audibility 
accepted  as  commercial  by  the  four  observers  named.  When 
the  cables  were  alone  in  circuit  some  of  the  observers  noticed 
that  in  the  case  of  the  new  cable  there  was  a  distinct  improvement 
in  the  quality  of  the  speech  as  compared  with  the  old  cable. 

"  The  employment  of  unloaded  800-lb.  copper  aerial  conductors, 
such  as  are  in  use  for  the  most  important  long-distance  trunk 
circuits  in  this  country,  will  render  it  possible  for  very  satisfactory 
conversations  to  take  place  from  call-boxes  between  centres  in 
England  and  on  the  Continent  when  the  added  distances  from 
the  ends  of  the  cable  do  not  exceed  1,700  miles;  that  is  to  say, 
with  land-lines  of  this  description  well-maintained  conversations 
between  London  and  Astrakhan  on  the  Caspian  Sea  would  be 
possible.  In  his  inaugural  address  to  the  Institution,1  Sir  John 
Gavey  included  a  table  of  equivalents  of  the  various  types  of 
unloaded  conductors.  It  may  be  assumed  that  in  practice  aerial 
conductors  of  the  smaller  gauges  can  be  improved  by  loading 
twofold,  and  the  conductors  in  cables  threefold,  so  that  it  is  not 
difficult  to  determine  the  centres  between  which  the  new  Anglo- 
French  telephone  cable  will  provide  communication,  assuming 
that  a  particular  type  of  conductor  is  employed  to  complete  the 
circuit." 

5.    Effect    of    Leakance    on     Loaded    Cables.— 

A  brief  reference  has  already  been  made  to  the  influence  of 
leakance  in  the  case  of  loaded  cables  upon  the  value  of  the 
attenuation  constant  in  connection  with  the  doubt  thrown  upon 
the  possibility  of  effectively  loading  gutta-percha  insulated  cables. 
This  question  is  important,  and  must  be  considered  a  little  more 
at  length.  It  has  been  dealt  with  in  a  paper  by  Dr.  A.  E.  Kennelly 
to  which  reference  has  already  been  made,  viz.,  "  On  the  Distri- 
bution of  Pressure  and  Current  over  Alternating  Current 
Circuits "  (see  Harvard  Engineering  Journal,  1905 — 1906), 
under  the  heading  "  Effect  of  Dielectric  Losses  on  Loading." 
Dr.  Kennelly  discusses  this  matter  as  follows : 

1  See  Sir  John  Gavey's  Inaugural  Address,  Journal  of  the  Institution  of 
Electrical  Engineers,  Vol.  XXXVI.,  p.  26,  1905. 


LOADED  CABLES  IN  PRACTICE       293 

Let  the  conductor  impedance  of  the  cable,  viz.,  the  quantity 
E  +jpL,  be  denoted  by  Zc  /  6C  as  a  vector.  Then,  equating 
the  sizes,  we  have 

Z^Rt+ptL*  and  tan  Oc=^. 

The   ratio  Lp/R    may  be  called  the   reactance  factor   of  the 
conductor  at  the  angular  velocity  p. 

Also  the  dielectric  admittance  of  the  cable,  viz.,  the  quantity 
$  +  JpVt  may  be  denoted  as  a  vector  by  YD  /  0D,  and  hence 


r^SH^C2  and  tan  0D  =       . 

£> 

The  ratio  of  the  susceptance  Cp  to  the  dielectric  conductance  S 
at  a  particular  angular  velocity  p  may  be  called  the  susceptance 
factor  of  the  cable,  although  cable  electricians  generally  deal 

more  with  the  quantity  -^  as  the  ratio  to  be  measured.      In  any 

case  -£•  is  the  tangent  of  the  angle  of  slope  of  the  vector  YD. 

Loading  a  circuit  obviously  increases  the  slope  of  the  vector 
impedance  Zc.  This  is  particularly  noticed  in  the  case  of 
telephone  cables,  in  which  when  unloaded  the  reactance  factor 

-£-  at  a  frequency  of  800  or  for  p  —  5,000  may  be  of  the  order  of 

0'03  to  0'05,  and  the  vectorial  angle  6C  may  be  1°  30'  or  2°'0  or 
so.  On  the  other  hand,  if  there  is  no  dielectric  loss  S  is  zero, 
and  the  slope  of  the  admittance  vector  is  90°,  since  then  its 
tangent  Cp/S  is  infinite.  In  such  cases  we  may  theoretically 
diminish  the  attenuation  constant  without  limit  by  increasing 
the  inductance  of  the  line  per  unit  of  length.  For  the  attenuation 
constant  a  is  the  real  part  of  the  product  of  ^R-\-jpL±  and 


+  jpC.  The  reader  should  remember  that  to  square-root  a 
vector  we  have  to  square-root  its  size  and  reduce  the  slope  to 
half,  whilst  to  obtain  the  product  of  two  vectors  we  have  to 
multiply  the  sizes  and  add  the  slopes.  Hence,  leaving  out  of 
account  sizes,  we  may  say  that  if  L  and  S  are  both  very  small, 
then  the  slope  of  the  conductor  impedance  vector  is  nearly  zero, 
and  that  of  the  dielectric  admittance  vector  is  nearly  90°.  Hence 
the  vector  representing  the  square  root  of  their  product,  or  the 


294        PEOPAGATION   OF   ELECTRIC  CURRENTS 

propagation  constant,  has  a  slope  of  45°.  If  we  keep  S  small,  but 
make  L  very  large,  then  the  slope  of  both  impedance  and 
admittance  vectors  is  nearly  90°,  and  the  square  root  of  their 
product,  or  the  propagation  constant,  has  also  aslope  of  nearly  90°. 
Hence  its  horizontal  step,  or  real  part  which  is  the  attenuation 
constant,  will  be  small.  If,  however,  S  is  large,  the  slope  of  the 
admittance  vector  is  much  less  than  90°  and  that  of  its  square 
root  much  less  than  45°,  and  hence  even  if  the  slope  of  the 
impedance  vector  is  90°  the  slope  of  the  propagation  constant  is 
something  considerably  less  than  90°,  and  that  means  that  the 
attenuation  constant  cannot  be  reduced  to  zero.  In  fact,  if  S  is 
not  zero,  but  has  an  appreciable  value,  then  it  is  useless  to  load 
the  cable  beyond  the  point  at  which  Lp/R  becomes  equal  to 
Cp/S.  For  the  attenuation  constant 


and  if  we  consider  7i,  S,  C,  and  p  to  be  constant  and  L  variable 
it  is  very  easy  to  prove  in  the  ordinary  way  by  finding  the 

differential     coefficient  -£=-  and    equating   it   to   zero    that   the 

/nriD 

above  expression  for  a  has  a  minimum  value  when  L  =  -n—  , 
in  other  words  when  -^=-5?,  that  is  when  Oc  =  0D,  or  when 

£1  O 

the  cable  is  distorsionless.  If  then  there  is  sensible  leakance  in 
the  dielectric  the  attenuation  constant  a  cannot  be  reduced  below 
the  value  a  =VSE  which  it  has  when  the  cable  fulfils  the 
Heaviside  conditions,  L/E  =  C/S,  for  being  distorsionless.  It 
follows  then  that  in  the  case  of  loaded  cables  great  care  must  be 
taken  to  keep  the  leakance  S  very  small,  or  nearly  zero.  This 
accounts  for  part  of  the  difficulty  of  loading  aerial  lines. 

If  we  write  down  the  already-given  formula  for  the  attenuation 
constant  a  of  a  cable,  viz., 


it  is  easily  transformed  into 


If  then       =P>  we  have  a 


LOADED  CABLES  IN  PKACTICE       295 

If  S  is  absolutely  zero,  then  by  making  pL  or  L  sufficiently 
large  compared  with  R  we  can  reduce  the  value  of  a  indefinitely. 
But  if  S  has  a  finite  value,  then  beyond  a  certain  point,  viz., 

Q 

when  L  =  R-^,  we  do  not  decrease,  but  actually  increase,  the 

value  of  a. 

Accordingly,  although  in  perfectly  insulated  lines  we  may 
with  advantage  increase  almost  indefinitely  the  inductance, 
provided  we  do  not  increase  the  resistance  at  the  same  time; 
yet  in  imperfectly  insulated  lines  there  is  a  limit  beyond  which 
increase  of  the  inductance  increases  instead  of  diminishing  the 
attenuation  constant. 

The  table  on  p.  296,  taken  from  Dr.  Kennelly's  paper  on  "  The 
Distribution  of  Pressure  and  Current  over  Alternating  Current 
Circuits,"  shows  the  difference  produced  in  loading  a  line  of  abso- 
lutely zero  leakance  up  to  200  niillihenrys  per  kilometre  and  the 
same  loading  for  a  line  having  an  insulation  resistance  of  10,000 
ohms  per  kilometre,  or  a  leakance  of  10~4  mhos  per  kilometre. 
In  the  first  case  the  loading  produces  a  remarkable  reduction 
in  the  attenuation  constant,  and  in  the  second  case  it  produces 
very  little. 

It  is  abundantly  clear,  therefore,  that  a  loaded  cable  must  be 
a  well-insulated  cable  if  we  are  to  obtain  the  benefit  of  the  loading 
in  the  form  of  a  small  attenuation  constant. 

It  is  this  fact,  combined  with  the  large  dielectric  current 
of  gutta-percha-covered  cable,  which  threw  doubt  originally 
upon  the  possibility  of  effectively  loading  submarine  telephone 
cables  insulated  with  G.P,  But  these  doubts  have  been  re- 
moved by  the  success  of  the  1910  Anglo-French  Channel 
telephone  cable. 

It  is,  however,  essential  to  secure  good  insulation  for  the 
loading  coils  themselves  in  underground  telephone  cables.  The 
practice  of  the  National  Telephone  Company  in  this  matter  is  to 
build  underground  pits  at  regular  intervals  of  a  mile  or  two,  as 
the  case  may  be,  and  place  in  these  cast-iron  watertight  boxes  in 
which  are  contained  the  highly  insulated  loading  coils. 

The  lead-covered  paper-insulated  cable  enclosing  many  strands 
or  separate  pairs  of  conductors  passes  through  this  pit  (see 


296       PKOPAGATION  OF  ELECTEIC  CURRENTS 


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LOADED  CABLES  IN  PRACTICE  297 

Fig.  8),  and  the  coils  are  connected  into  the  different  circuits. 
In  this  manner  good  insulation  is  secured  for  the  line  and  coils. 

The  attenuation  constant  of   the  loaded  line  can  always  be 
calculated  very  approximately  by  the  formula 


This  formula  is  arrived  at  in  the  following  manner  : 
By  the  binomial  theorem    we  have  for  the  expansion  of  a 
binomial  (a  +  n)n  the  series 


If  n  =    ,  then 


-1  x-\  —  ^    ~"    an~2  n2+etc. 


--—-{-etc. 


2    a 
Hence  if  x  is  small   compared  with  a,  so  that  we  can  neglect 

_  _  SY* 

powers  of  x/a,  we  have  v  a  +  x  =  v  a  +  ;—  /^  nearly. 

Accordingly,  if  R  is  small  compared  with  pL  and  S  is  small 
compared  with  pC,  we  have 


_  o     o 

and  VS2+/2C2=jpC+2^. 

Since,  then,  2«2  =  VR*+p*L*VS*+p*C+SR-p*LC,  it  follows 
that  when  R/pL  and  S/pC  are  both  small  quantities  compared 
with  unity  we  have 


or  a  =  • 

Accordingly  the  attenuation   is    greatly  affected  by  the  value 
of  SIC. 

No  really  satisfactory  method  has  yet  been  found  for  measuring 
the  value  of  the  leakance  S  or  the  ratio  S/C  for  telephonic 
frequencies,  but  it  is  found  that  by  taking  S/C=8Q  this  formula 
gives  attenuation  constants  which  are  in  close  agreement  with 


298        PEOPAGATION   OF   ELECTKIC   CURRENTS 


observed  values  for  loaded  cables.  Thus,  in  a  discussion  on  a 
paper  by  Professor  Perry  on  "  Telephone  Circuits,"  Mr.  A.  W. 
Martin,  of  the  General  Post  Office,  gave  some  useful  measure- 
ments confirming  this  result  for  loaded  cables. 

Cables  of  various  lengths  were  loaded  with  iron-cored  inductance 
coils,  each  having  effective  resistances  of  5'4  ohms  at  750  fre- 
quency and  15'0  ohms  at  2,000  and  3'5  ohms  for  steady  currents, 
also  an  inductance  of  0'135  henry  per  coil.  These  coils  were 
inserted  at  various  intervals  in  a  line  of  conductor  resistance 
18  ohms  per  mile  of  loop,  and  capacity  0'055  m.f.d.,  and  induct- 
ance 0*001  henry  per  mile  of  loop.  The  attenuation  constants 
were  then  calculated  from  the  above  formula,  taking  S/C  =  80, 
and  they  were  also  measured,  and  the  results  were  as  follows : 


Interval 
between 

Attenuation  Constants  for 
Frequency  750. 

Coils  pei- 
Wave  at  a 

Loading  Coils 

Frequency 

Articulation. 

in  miles. 

Calculated. 

Observed. 

of  2,000. 

1-1 

0-011 

0-013 

5-6 

Very  good 

2-1 

0-012 

0-012 

4-0 

Very  good 

3-2 

0-013 

0-012 

3-3 

Good 

4-3 

0-014 

0-014 

2-8 

Bad 

Unloaded 

0-042 

0-045 

— 

— 

In  the  case  of  the  Anglo-French  telephone  cable  (1910)  above 
described,  the  observed  attenuation  constant  corresponds  to  a 
value  of  SIC  =  99  instead  of  80.  There  is  no  doubt  that  the 
ratio  of  S/C  for  any  telephone  conductor  plays  a  very  important 
part  in  determining  the  speech- transmitting  efficiency. 

In  the  United  States  one  of  the  principal  difficulties  in  con- 
nection with  the  loading  of  long  distance  aerial  telephone  lines 
has  been  the  leakage  over  the  insulators,  and  a  more  efficient  type 
of  glass  insulator  has  had  to  be  substituted  for  the  ordinary  type 
in  order  to  keep  down  the  leakage,  which  prevents  the  loading 
from  having  its  full  effect. 

The  reader  will  find  a  considerable  amount  of  valuable  infor- 
mation on  the  properties  of  loaded  lines  in  the  discussion  which 


LOADED  CABLES  IN  PRACTICE       299 

took  place  at  the  Physical  Society  of  London  on  a  paper  by 
Professor  Perry  in  1910  (see  The  Electrician,  March  llth,  1910, 
p.  879),  and  also  a  longer  and  even  more  important  discussion 
which  took  place  at  the  Institution  of  Electrical  Engineers  on 
the  paper  by  Major  O'Meara  on  "  Submarine  Cables  for  Long 
Distance  Telephone  Circuits"  (see  The  Electrician,  Vol.  LXV., 
p.  609,  1910,  and  Vol.'LXVL,  pp.  375,  417,  419,  589,  and  615, 
1911),  in  which  all  the  leading  experts  in  telephony  and 
telegraphy  in  England  took  part. 


APPENDIX. 


The  table  below  is  taken  by  kind  permission  from  a  paper  by 
Dr.  A.  E.  Kennelly,  published  in  the  Harvard  Engineering 
Journal,  May,  1903. 

TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OP  HYPERBOLIC  ANGLES. 

The  Sines,  Cosines,  and  Tangents  have  been  taken  from  Ligowski's  Tables 
published  in  Berlin  in  1890.  The  Cotangents,  Secants,  and  Cosecants  have  been 
deduced  from  the  preceding  quantities. 


H. 

Sinh.  u. 

Cosh.  u. 

Tanh.  u. 

Coth.  u. 

Sech.  u. 

Cosech.  u. 

u. 

000 

o- 

1-000 

o- 

00 

1-00 

00 

000 

o-oi 

0-02 

0-03 

o-oioouo 

0-020001 
0-030005 

1-000050 
1-000200 
1  -000450 

o-oiooo 

0-02000 
0-02999 

100- 
50- 
33-34 

0-9999 
0-9998 
0-9995 

100- 
50- 
33-333 

o-oi 

0-02 
0-03 

0-04 

()•():, 
0-06 

o-oiooil 
0-050021 
0-060036 

1-000800 
1-001250 
1-001801 

0-03998 
0-04996 
0-05993 

25-013 
20-016 
16-686 

0-9992 
0-9987 
0-9982 

24-99 
19-992 

16-657 

0-04 

0-05 
0-06 

0-07 
0-08 
0-09 

0-070057 
0-080085 
0-090122 

1-002451 
1-003202 
1-004053 

0-06989 
0-07983 
0-08976 

14-308 
12-527 
11-141 

0-9975 
0-9968 
0-9959 

14-274 
12-487 
11-097 

0-07 
0-08 
0-09 

010 

0-100167 

1-005004 

0-09967 

10-033 

0-9950 

9-983 

010 

0-11 

0-12 
0-13 

0-110222 
0-120288 
0-1303U6 

1-006056 
1-007209 
1-008462 

0-10956 
0-11943 
0-12927 

9-128 
8-373 
7-735 

0-9940 
0-9928 
0-9916 

9-073 
8-314 
7*669 

0-11 
0-12 
0-13 

0-14 
0-15 
0-16 

0-140458 
0-150563 
0-160684 

1-009816 
1-011271 
1-012827 

0-13909 
0-14888 
0-15865 

7-189 
6-716 
6-303 

0-9902 
0-9888 
0-9873 

7-120 
6-642 
6-223 

o-ll 
0-15 
0-16 

0-17 
0-18 
0-19 

0-170820 
0-180974 
0-191145 

1-014485 
1-016244 
1-018104 

0-16838 
0-17808 
0-18775 

5-939 
5-615 
5-325 

0-9857 
0-9840 

0-9822 

5-854 
5-525 
5-232 

0-17 
0-18 
0-19 

302 


APPENDIX 


TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OP  HYPERBOLIC  ANGLES.— continued. 


u. 

Sinh.  u. 

Cosh.  u. 

Tanh.  u. 

Coth.  u. 

Sech.  u. 

Cosech.  n. 

u. 

020 

0-201336 

1-020067 

0-19737 

5-067 

0-9803 

4-967 

020 

0-21 
0-22 
0-23 

0-211547 
0-221779 
0-232033 

1-022131 
1-024298 
1-026567 

0-20696 
0-21652 
0-22603 

4-832 
4-618 
4-425 

0-9784 
0-9763 
0-9742 

4-726 
4-509 
4-310 

0-21 
0-22 
0-23 

0-24 
0-25 
0-26 

0-242311 
0-252612 
0-262939 

1-028939 
1-031413 
1-033991 

0-23549 
0-24492 
0-25430 

4-246 
4-083 
3-932 

0-9719 
0-9695 
0-9671 

4-127 
3-959 
3-803 

0-24 
0-25 
0-26 

0-27 
0-28 
0-29 

0-273292 
0-283673 
0-294082 

1-036672 
1-039457 
1-042346 

0-26363 
0-27290 
0-28214 

3-793 
3-664 
3-544 

0-9046 

0-9620 
0-9591 

3-659 
3-525 
3-400 

0-27 
0-28 
0-29 

030 

0-304520 

1-045339 

0-29131 

3-433 

0-9566 

3-284 

030 

0-31 
0-32 
0-33 

0-314989 
0-325489 
0-336022 

1-048436 
1-051638 
1-054946 

0-30043 
0-30951 
0-31852 

3-328 
3-231 
3-140 

0-9537 
0-9511 
0-9479 

3-175 
3-072 
2-976 

0-31 
0-32 
0-33 

0-34 
0-35 
0-36 

0-346589 
0-357190 
0-367827 

1-058359 

1-061878 
1-065503 

0-32748 
0-33637 
0-34522 

3-053 
2-973 

2-897 

0-9447 
0-9416 
0-9385 

2-885 
2-800 
2-719 

0-34 
0-35 
0-36 

0-37 
0-38 
0-39 

0-378500 
0-389212 
0-399902 

1-069234 
1-073073 
1-077019 

0-35399 
0-36271 
0-37136 

2-825 
2-757 
2-693 

0-9353 
0-9319 
0-9285 

2>642 

2-569 
2-500 

0-37 
0-38 
0-39 

0-40 

0-410752 

1-081072 

0-37995 

2-632 

0-9250 

2-434 

040 

0-41 
0-42 
0-43 

0-421584 
0-432457 
0-443374 

1-085234 
1-089504 
1-093883 

0-38847 
0-39693 
0-40532 

2-574 
2-512 
2-467 

0-9215 
0-9178 
0-9141 

2-372 
2-312 

2-256 

0-41 
0-42 
0-43 

0-44 
0-45 
0-46 

0-454335 
0-465342 
0-476395 

1-098372 
•102970 
•107679 

0-41365 
0-42190 
0-43009 

2-417 
2-370 
2-325 

0-9103 
0-9066 
0-9025 

2-201 
2-149 
2-099 

0-44 
0-45 
0-46 

0-47 
0-48 
0-49 

0-487496 
0-498646 
0-509845 

•112498 
•117429 
•122471 

0-43820 
0-44624 
0-45421 

2-282 
2-241 
2-202 

0-8988 
0-8949 
•  0-8909 

2-051 
2-006 
1-961 

0-47 
0-48 
0-49 

050 

0-521095 

1-127626 

0-46211 

2-164 

0-8868 

1-919 

050 

0-51 
0-52 
0-53 

0-532398 
0-543754 
0-555164 

1-132893 
1-138274 
1-143769 

0-46995 
0-47769 
0-48538 

2-128 
2-093 
2-060 

0-8827 
0-8785 
0-8743 

1-878 
1-839 
1-801 

0-51 
0-52 
0-53 

0-54 
0-55 
0-56 

0-566629 
0-578152 
0-589732 

1-149378 
1-155101 
1-160941 

0-49299 

0-50052 
0-50797 

2-028 
1-998 
1-969 

0-8700 
0-8658 
0-8614 

1-765 
1-730 
1-696 

0-54 
0-55 
0-56 

0-57 
0-58 
0-59 

0-601371 
0-613070 
0-624831 

1-166896 
1-172968 
1-179158 

0-51536 
0-52266 
0-52990 

1-940 
1-913 

1-887 

0-8570 
0-8525 

0-8480 

1-663 
1-631 
1-601 

0-57 
0-58 
0-59 

APPENDIX 


303 


TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OF  HYPERBOLIC  ANGLES.—  continued. 


u. 

Sinh.  w. 

Cosh.  u. 

Tanh.  u. 

Coth.  u. 

Sech.  w. 

Cosech.  u. 

u. 

060 

0-636651 

1-185465 

0-53704 

1-862 

0-8435 

1-571 

060 

0*61 
0-62 
0-63 

0-648540 
0-660492 
0-672509 

1-191891 
1-198436 
1-205101 

0-54413 
0-55112 

9-55805 

1-838 
1-814 
1-792 

0*8390 
0*8344 
0-8298 

1-542 
1-514 
1-487 

0-61 

0*62 
0-63 

IN;  I 
0-65 
6-66 

0-684594 
0-696748 

0-708D70 

1-211887 
1-218793 
1-225882 

0-56490 
0-57166 
0-57836 

1-770 
1-749 
1-729 

0-8251 
0-8205 
0*8158 

1-461 
1-435 
1-410 

0-64 
0-65 
0-66 

0-67 
0-68 

0-69 

0-721264 
0-733630 
0-746070 

2-232973 
1-240247 
1-247646 

0-58498 
0-59152 
0-59798~ 

1-709 
1-690 
1-672 

0*8110 
0*8065 
0*8015 

1-387 
1-363 
1-340 

0*67 
0-68 
0-69 

0-70 

0-758584 

1-255169 

0-60437 

1*655 

0*7967 

1-318 

070 

071 
0-72 
0-73 

0-771171 

0-783840 
0-796586 

1-262818 
1-270593 
1-278495 

0-61067 
0-61691 
0-62306 

1*637 
1*621 
1*605 

0*7919 

0*7870 
0*7821 

1-297 
1-276 
1-255 

0*71 
0-72 
0-73 

0-74 

0-75 
0-76 

0-809411 
0-822317 
0-835305 

1-286525 
1-294683 
1-302971 

0-62914 
0-63516 
0-64K-8 

1*590 
1*574 
1*5599 

0*7773 
0*7724 

0-7675 

1-235 
1-216 
1-1972 

0-74 
0-75 
0-76 

0-77 
0-78 
0-79 

0-848377 
0-861533 
0-874776 

1-311390 
1-319939 
1-328621 

0-64693 
0-65271 
0-65842 

1*5457 
1*5320 

1*5188 

0-7625 
0-7576 
0-7527 

1-1787 
1-1607 
1-1431 

0-77 
0-78 
0-79 

080 

0-888106 

1-337435 

0-66403 

1-5059 

0-7477 

1-1259 

080 

0-81 
0-82 
0-83 

0-901525 
0-915034 
9-928635 

1-346383 

1-3554C.C. 
1-364684 

0-66959 
0-67507 
0-68047 

1-4934 
1-4813 
L-4696 

0-7427 
0-7377 
0-7327 

1-1092 
1-0928 
1*0768 

0-81 
0-82 
0-83 

0-84 
0-85 
0-86 

0-942328 
0-956116 
0-969999 

0-374039 
1-383531 
1-393161 

0-68580 
0-69107 
0-69626 

1-4582 
1*4470 
1*4362 

0-7278 
0-7228 
0-7178 

1*0612 
1*0459 
1*0309 

0-84 
0-85 
0-86 

0-87 
0-88 
0-89 

0-983980 
0-998058 
1-012237 

1-402931 
1-412841 
1-422893 

0-70137 
0-70642 
0-71139 

1*4258 
1*4156 
1-4057 

0-7128 
0-7078 
0-7028 

1*0163 
1-0020 
0-9881 

0-87 
0-88 
089 

090 

1-026517 

1-433086 

0-71629 

1-3961 

0-6978 

0-9737 

090 

0-91 
0-92 
0-93 

1-040899 
1-055386 
1-069978 

4-443423 

1-45390.", 
1-464531 

0-72114 
0-72591 
0-73060 

1-3867 
1-3776 
1-3687 

0-6928 
0-6878 
0-6828 

0-9607 
0-9475 
0-9346 

0-91 

0*92 
0*93 

0-94 
0-95 
0-96 

1-084677 
1-099484 
1-114402 

1-475305 
1-486225 
1-497295 

0-73522 
0-73979 
0-74427 

1-3600 
1-3517 
1-3436 

0-6778 
0-6728 
0-6678 

0-9219 
0-9095 
0-8973 

0*94 
0*95 
0-96 

0-97 
0-98 
0-99 

1-129431 
1-144573 
1-159829 

1-508514 
1-519884 
1-531406 

0-74870 
0-75306 
0-75736 

1-3356 
1-3279 
1-3204 

0-6629 
0-6579 
0-6529 

0-8854 
0-8737 
0*8621 

0-97 

0-98 
0-99 

304 


APPENDIX 


TABLE  OP  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OF  HYPERBOLIC  ANGLES.— continued. 


u. 

Sinh.  u. 

Cosh.  ?/. 

Tanh.  u. 

'  Coth.  u. 

Sech.  u. 

Cosech.  u. 

u. 

100 

1-175201 

1-543081 

0-76159 

1-3130 

0-6480 

0-8509 

100 

1-01 

1-190691 

1-554910 

0-76576 

1-3059 

0-6431 

0-8395 

1-01 

1-02 

1-206300 

1-566895 

0-76987 

1-2989 

0-6382 

0-8290 

1-02 

1-03 

1-222029 

1-579036 

0-77391 

1-2921 

0-6333 

0-8183 

1-03 

1-04 

1-237881 

1-591336 

0-77789 

1-2855 

0-6284 

0-8078 

1-04 

1-05 

1-253857 

1-603794 

0-78181 

1-2791 

0-6235 

0-7975 

1-05 

1-06 

1-269958 

1-616413 

0-78566 

1-2728 

0-6186 

0-7874 

1-06 

1-07 

1-286185 

1-629194 

0-78846 

1-2666 

0-6138 

0-7777 

1-07 

1-08 

1-302542 

1-642138 

0-79320 

1-2607 

0-6090 

0-7677 

1-08 

1-09 

1-319029 

1-655245 

0-79688 

1-2549 

0-6042 

0-7581 

1-09 

1-10 

1-335647 

1-668519 

0-80050 

1-2492 

0-5993 

0-7487 

1-10 

1-11 

1-352400 

1-681959 

0-80406 

1-2437 

0-5945 

0-7393 

1-11 

1-12 

1-369287 

1-695567 

0-80757 

1-2382 

0-5898 

0-7302 

1-12 

1-13 

1-386312 

1-709345 

0-81102 

1-2330 

0-5850 

0-7215 

1-13 

1-14 

1-403475 

1-723294 

0-81441 

1-2279 

0-5803 

0-7125 

1-14 

1-15 

1-420778 

1-737415 

0-81775 

1-2229 

0-5755 

0-7038 

1-15 

1-16 

1-438224 

1-751710 

0-82104 

1-2180 

0-5708 

0-6953 

1-16 

1-17 

1-455813 

1-766180 

0-82427 

1-2132 

0-5662 

0-6869 

1-17 

1-18 

1-473548 

1-780826 

0-82745 

1-2085 

0-5616 

0-6786 

1-18 

1-19 

1-491430 

1-795651 

0-83058 

1-2040 

0-5569 

0-6705 

1-19 

1-20 

1-509461 

1-810656 

0-83365 

1-1995 

0-5523 

0-6625 

120 

1-21 

1-527644 

1-825841 

0-83668 

1-1952 

0-5477 

0-6546 

1-21 

1-22 

1-545979 

1-841209 

0-83965 

1-1910 

0-5431 

0-6468 

1-22 

1-23 

1-564468 

1-856761 

0-84258 

1-1868 

0-5385 

0-6392 

1-23 

1-24 

1-583115 

1-872499 

0-84546 

•1828 

0-5340 

0-6317 

1-24 

1-25 

1-601919 

1-888424 

0-84828 

•1789 

0-5296 

0-6242 

1-25 

1-26 

.  1-620884 

1-904538 

0-85106 

•1750 

0-5251 

0-6170 

1-26 

1-27 

1-640010 

1-920842 

0-85380 

•1712 

0-5206 

0-6098 

1-27 

1-28 

1-659301 

1-937339 

0-85648 

•1675 

0-5162 

0-6026 

1-28 

1-29 

1-678758 

1-954029 

0-85913 

1-1640 

0-5118 

0-5957 

1-29 

1-30 

1-698382 

1-970914 

0-86172 

1-1604 

0-5074 

0-5888 

1-30 

1-31 

1-718177 

1-987997 

0-86428 

1-1570 

0-5030 

0-5820 

1-31 

1-32 

1-738143 

2-005278 

0-86678 

1-1537 

0-4987 

0-5753 

1-32 

1-33 

1-758283 

2-022760 

0-86925 

1-1504 

0-4944 

0-5687 

1-33 

1-34 

1-778599 

2-040445 

0-87167 

1-1472 

0-4901 

0-5623 

1-34 

1-35 

1-799093 

2-058333 

0-87405 

1-1441 

0-4858 

0*5559 

1-35 

1-36 

1-819766 

2-076427 

0-87639 

1-1410 

0-4816 

0-5495 

1-36 

1-37 

1-840622 

2-094729 

0-87869 

1-1380 

0-4773 

0-5433 

1-37 

1-38 

4-861662 

2-113240 

0-88095 

1-1351 

0-4732 

0-5372 

1-38 

1-39 

1-882887 

2-131963 

0-88317 

1-1323 

0-4690 

0-5311 

1-39 

APPENDIX 


305 


TABLE  OP  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OF  HYPERBOLIC  ANGLES.— continued. 


u. 

Smb.  u. 

Cosh.  u. 

Tanb.  u. 

Coth.  u. 

Sech.  u. 

Cosecb.K. 

a. 

1-40 

1-904302 

2-150898 

0-88535 

1-1295 

0-4649 

0-5252 

1-40 

1-41 
1-42 
1-43 

1-925906 
1-947703 
1-9G9695 

2-170049 
2-189417 
2-209004 

0-88749 
0-88960 
0-89167 

1-1268 
1-1241 
1-1215 

0-4608 
0-4568 
0-4527 

0-5192 
0-5134 
0-5077 

1-41 
1-42 
1-43 

1-44 
1-46 

1-46 

1-991884 
2-014272 
2-036862 

3-228812 
2-248842 
2-269098 

0-89370 
0-89569 
0-89765 

1-1189 
1-1165 
1-1140 

0-4486 
0-4446 
0-4407 

0-5020 
0-4964 
0-4909 

1-44 
1-45 
1-46 

1-47 
1-48 
1-49 

2-059655 
2-082654 
2-105861 

2-289580 
2-310292 
2-331234 

0-89958 
0-90147 
0-90332 

1-1116 
1-1093 
1-1070 

0-4367 
0-4329 
0-4290 

0-4855 
0-4802 
0-4749 

1-47 
1-48 
1-49 

150 

2-129279 

2-352410 

0-90515 

1-1048 

0-4251 

0-4697 

1-50 

1-51 
1-52 
1-53 

2-152910 
2-176757 

2-200821 

1-373820 
2-395469 
2-417356 

0-90694 
0-90870 
0-91042 

1-1026 
1-1005 
1-0984 

0-4212 
0-4174 
0-4137 

0-4645 
0-4594 
0-4543 

1-51 
1-52 
1-53 

1-54 
1*55 
1-56 

2-225105 
2-249611 
2-274343 

2-439486 
2-461859 
2-484479 

0-91212 
0-91379 
0-91542 

1-0963 
1-0943 
1-0924 

0-4099 
0-4062 
0-4025 

0-4494 
0-4444 

0-4398 

1-54 
1-55 
1-56 

1-57 
1-58 
1-69 

2-299302 
2-324490 
2-349912 

2-507347 
2-530465 
2-553837 

0-91703 
0-91860 
0-92015 

1-0905 

1-0886 
1-0868 

0-3988 
0-3952 
0-3916 

0-4350 
0-4302 
0-4255 

1-57 
1-58 
1-59 

160 

2'375568 

2-577464 

0-92167 

1-0850 

0-3879 

0-4209 

1-60 

1-61 
1-62 
1-63 

2-401462 
2-427596 
2-453973 

2-601349 
2-625495 
2-649902 

0-92316 
0-92462 
0-92606 

1-0832 
1-0815 
1-0798 

0-3844 
0-3809 
0-3774 

0-4164 
0-4119 
0-4075 

1-61 
1-62 
1-63 

1-64 
1-65 
1-66 

2-480595 
2-507465 
2-534586 

2-674575 
2-699515 
2-724725 

0-92747 
0-92886 
0-93022 

1-0782 
1-0765 
1-0750 

0-3739 
0-3704 
0-3670 

0-4031 
0-3988 
0-3945 

1-64 
1-65 
1-66 

1-67 
1-68 
1-69 

2-561960 

2-589591 
2-617481 

2-750207 
2-775965 
2-802000 

0-93155 
0-93286 
0-93415 

1-0735 
1-0719 
1-0704 

3-3636 
0-3602 
0-3569 

0-3903 
0-3862 
0-3820 

1-67 
1-68 
1-69 

1-70 

2-645632 

2-828315 

0-93541 

1-0690 

0-3536 

0-3780 

170 

1-71 
1-72 
1-73 

2-674048 
2-702731 
2-731685 

2-854914 
2-891797 
2-908969 

0-93665 
0-93786 
0-93906 

1-0676 
1-0662 
1-0649 

0-3503 
0-3470 
0-3438 

0-3740 
0-3700 
0-3661 

1-71 
1-72 
1-73 

1-74 
1-75 
1-76 

2-760912 
2-790414 
2-820196 

2-936432 
2-964188 
2-992241 

0-94023 
0-94138 
0-94250 

1-0636 
1-0623 
1-0610 

0-3405 
0-3373 
0-3342 

0-3622 
0-3584 
0-3546 

1-74 
1-75 
1-76 

1-77 

1-78 
1-79 

2-850260 
2-880609 
2-911246 

3-020593 
3-049247 
3-078206 

0-94361 
0-94470 
0-94576 

1-0597 
1-0585 
1-0573 

0-3310 
0-3279 
0-3248 

0-3508 
0-3471 
0-3435 

1-77 
1-78 
1-79 

B.C. 


306 


APPENDIX 


TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OP  HYPERBOLIC  ANGLES. — continued. 


u. 

Sinh.  u. 

,Cosh.  u. 

Tanh.  u. 

Coth.  u. 

Sech.  «. 

Cosech.w. 

». 

1-80 

2-942174 

3-107473 

0-94681 

1-0561 

0-3218 

0-3399 

1-80 

1-81 
1-82 
1-83 

2-973397 
3-004916 
3-036737 

3-137051 
3-166942 
3-197150 

0-94783 
0-94884 
0-94983 

1-0550 
1-0539 
1-0528 

0-3187 
0-3158 
0-3128 

0-3363 
0-3328 
0-3293 

1-81 
1-82 
1-83 

1-84 
1-85 
1-86 

3-068860 
3-101291 
3-134032 

3-227678 
3-258528 
3-289705 

0-95080 
0-95175 
0-95268 

1-0517 
1-0507 
1-0497 

0-3098 
0-3069 
0-3040 

0-3258 
0-3224 
0-3191 

1-84 
1-85 
1-86 

1-87 
1-88 
1-89 

3-167086 
3-200457 
3-234148 

3-321210 
3-353047 
3-385220 

0-95359 
0-95449 
0-95537 

1-0487 
1-0477 
1-0467 

0-3011 
0-2982 
0-2954 

0-3157 
0-3125 
0-3092 

1-87 
1-88 
1-89 

1-90 

3-268163 

3-417732 

0-95624 

1-0457 

0-2926 

0-3059 

1-90 

1-91 
1-92 
1-93 

3-302504 
3-337176 
3-372181 

3-450585 
3-483783 
3-517329 

0-95709 
0-95792 
0-95873 

1-0448 
1-0439 
1-0430 

0-2897 
0-2870 
0-2843 

0-3028 
0-2997 
0-2965 

1-91 
1-92 
1-93 

1-94 
1-95 
1-96 

3-407524 
3-443207 
3-479234 

3-551227 
3-585481 
3-620093 

0-95953 
0-96032 
0-96109 

1-0422 
1-0413 
1-0405 

0-2816 
0-2789 
0-2762 

0-2935 
0-2904 

0-2874 

1-94 
1-95 
1-96 

1-97 
1-98 
1-99 

3-515610 
3-552337 
3-589419 

3-655067 
3-690406 
3-726115 

0-96185 
0-96259 
0-96331 

1-0397 
1-0389 
1-0380 

0-2736 
0-2710 
0-2684 

0-2844 
0-2815 
0-2786 

1-97 
1-98 
1-99 

200 

3-626860 

3-762196 

0-96403 

1-0373 

0-2658 

0-2757 

200 

2-01 
2-02 
2-03 

3-66466 
3-70283 
3-74138 

3-79865 
3-83549 
3-87271 

0-96473 
0-96541 
0-96608 

1-0365 
1-0358 
1-0351 

0-2632 

0-2607 
0-2582 

0-2729 
0-2701 
0-2673 

2-01 
2-02 
2-03 

2-04 
2-05 
2-06 

3-78029 
3-81958 
3-85926 

3-91032 
3-94832 
3-98671 

0-96675 
0-96740 
0-96803 

1-0344 
1-0337 
1-0330 

0-2557 
0-2533 
0-2508 

0-2645 
0-2618 
0-2596 

2-04 
2-05 
2-06 

2-07 
2-08 
2-09 

3-89932 
3-93977 
3-98061 

4-02550 
4-06470 
4-10430 

0-96865 
0-96926 
0-969^6 

1-0323 
1-0317 
1-0310 

0-2484 
0-2460 
0-2436 

0-2565 
0-2538 
0-2512 

2-07 
2-08 
2-09 

2-10 

4-02186 

4-14431 

0-97045 

1-0304 

0-2413 

0-2486 

210 

2-11 
2-12 
2-13 

4-06350 
4-10555 
4-14801 

4-18474 
4-22558 

4-26685 

0-97101 
0-97159 
0-97215 

1-0298 
1-0293 

1-0286 

0-2389 
0-2366 
0-2344 

0-2461 
0-2436 
0-2411 

2-11 
2-12 
2-13 

2-14 
2-15 
2-16 

4-19089 
4-23419 
4-27791 

4-30855 
4-35067 
4-39323 

0-97274 
0-97323 
0-97375 

1-0280 
1-0275 
1-0269 

0-2321 
0-2298 
0-2276 

0-2386 
0-2362 
0-2338 

2-14 
2-15 
2-16 

2-17 
2-18 
2-19 

4-32205 
4-36663 
4-41165 

4-43623 
4-47967 
4-52356 

0-97426 
0-97477 
0-97524 

1-0264 
1-0259 
1-0254 

0-2254 
0-2232 
0-2211 

0-2314 
0-2290 
0-2267 

2-17 
2-18 
2-19 

APPENDIX 


307 


TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OP  HYPERBOLIC  ANGLES.— continued. 


u. 

Sinh.  a. 

Cosh.  «. 

Tanh.  u. 

Coth.  u. 

Sech.  «. 

Cosech.  u. 

u. 

220 

4-45711 

4-56791 

0-97574 

1-0249 

0-2189 

0-2244 

220 

2-21 
2-22 
2-23 

4-60301 
4-64936 
4-59617 

4-61271 
4-65797 
4-70370 

0-97622 
0-97668 
0-97714 

1-0243 
1-0239 
1-0234 

0-2168 
0-2147 
0-2126 

0-2221 
0-2198 
0-2176 

2-21 
2-22 
2-23 

2-24 
2-2S 

2-26 

4-04344 
4-<;<)117 
4-73937 

4-74989 
4-79657 
4-84372 

0-97758 
0-97803 
0-97847 

1-0229 
1-0224 
1-0220 

0-2105 
0-2085 
0-2064 

0-2154 
0-2132 
0-2110 

2-24 

2-25 
2-26 

2-27 
2-28 
2-29 

4-78804 
4-83720 
4-88683 

4-89136 
4-93948 
4-98810 

0-97888 
0-97929 
0-97970 

1-0216 
1-0211 
1-0207 

0-2044 
0-2024 
0-2005 

0-2089 
0-2067 
0-2047 

2-27 
2-28 
2-29 

230 

4-93696 

5-03722 

0-98010 

1-0203 

0-1985 

0-2026 

230 

2-31 
2-32 
2-33 

4-98758 
•Vi  >3870 
5-<  19032 

5-08684 
5-13697 
5-18762 

0-98049 
0-98087 
0-98124 

1-0199 
1-0195 
1-0191 

0-1966 
0-1947 
0-1928 

0-2006 

0-1985 
0-1965 

2-31 
2-32 
2-33 

2-34 
2-35 
2-36 

5-14245 
5-19510 
5-24827 

5-23879 
6-29047 

5-34269 

0-98161 
0-98198 
0-98233 

1-0187 
1-0183 
1-0180 

0-1909 
0-1890 
0-1872 

0-1945 
0-1925 
0-1905 

2-34 
2-35 
2-36 

2-37 
2-38 
2-39 

5-30196 

5-35618 
5-41093 

5-39544 

5-44873 
•V  50256 

0-98268 
0-98302 
0-98335 

1-0177 
1-0173 
1-0169 

0-1854 
0-1835 
0-1817 

0-1886 
0-1867 
0-1848 

2-37 
2-38 
2-39 

240 

.VIG623 

5-55695 

0-98368 

1-0166 

0-1800 

0-1829 

240 

2-41 
2-42 
2-43 

5-52207 
5-57847 
6-68642 

5-61189 
5-66739 
5-72346 

0-98399 
0-98431 
0-98462 

1-0163 
1-0159 
1-0156 

0-1782 
0-1766 

0-1747 

0-1811 
01793 
0-1775 

2-41 
2-42 
2-43 

2-44 

2-45 

2-46 

5-HD294 
."•75103 
5-80969 

5-78010 
5-83732 
5-89512 

0-98492 
0-98522 
0-98551 

1-0153 
1-0150 
1-0147 

0-1730 
0-1713 
0-1696 

0-1757 
0-1739 
0-1721 

2-44 
2-45 
2-46 

2-47 
2-48 
2-49 

5-86893 
5-92876 
5-98918 

5-95352 
6-01250 
6-07209 

0-9857.9 
0-98607 
0-98635 

1-0144 
1-0141 
1-0138 

0-1680 
0-1663 
0-1647 

0-1704 
0-1687 
0-1670 

2-47 
2-48 
2-49 

250 

6-05020 

6-13229 

0-98661 

1-0135 

0-1631 

0-1653 

250 

26 

6-69473 

6-76901 

0-98403 

1-0110 

0-1477 

0-1494 

26 

27 
28 
29 

7-40626 
8-19192 
9-05956 

7-47347 
8-25273 
9-11458 

0-99101 
0-99263 
0-99396 

1-0091 
1-0074 

1-0060 

0-1338 
0-1212 

0-1097 

0-1350 
0-1221 
0-1104 

27 
28 
29 

3-0 

10-01787 

10-06766 

0-99505 

1-0050 

0-0937 

0-09982 

30 

808 


APPENDIX 


TABLE  OF  SINKS.  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OF  HYPERBOLIC  ANGLES.— continued. 


11. 

Sirih.  u. 

Cosh.  «. 

Tanh.  u. 

Cotli.  u. 

Sech.  u. 

Cosech.  u. 

u. 

3-1 
32 
33 

11-07(545 
12-24588 
13-53788 

11-12150 
12-28665 
13-57476 

0-99595 
0-99668 
0-99728 

1-0041 
1-0033 
1-0027 

0-0899 
0-0814 
0-0736 

0-0903 
0-0816 
0-0739 

3-1 
32 
3-3 

3-4 
35 
3-6 

14-9(5536 
16-542(53 

1  8-2854(5 

14-99874 
16-57282 
18-31278 

0-99778 
0-99818 
0-99851 

1-0022 
1-0018 
1-0015 

0-0667 
0-0604 

0-0646 

0-0668 
0-0(504 
0-0547 

3-4 
35 
3-6 

3-7 
38 
3-9 

20-21129 
22-33941 
24-69110 

20-23601 
22-36178 
24-71135 

0-99878 
0-99900 
0-99918 

1-0012 
1-0010 

1-0008 

0-0494 
0-0447 
0-0405 

0-0495 
0-0448 
0-0405 

3-7 
3-8 
39 

40 

27-28992 

27-30823 

0-99933 

1-0007 

0-0366 

0-0366 

40 

4-1 
42 
43 

30-16186 
33-33567 
36-84311 

30-17843 
33-35066 

36-85668 

0-99945 
0-99955 
0-99963 

1-0006 
1-0005 
1-0004 

0-0331 
0-0300 
0-0271 

0-0332 
0-0300 
0-0271 

4-1 
4-2 
4-3 

4-4 
45 
46 

40-71930 
45-00301 
49-73713 

40-73157 
45-01412 
49-74718 

0-99970 
0-99975 
0-99980 

1-0003 
1-0003 
1-0002 

0-0245 
0-0222 
0-0201 

0-0245 
0-0222 
0-0201 

4-4 
4-5 
46 

4-7 
48 
49 

54-96904 
60-75109 
67-14117 

54-97813 
60-75932 
67-14861 

0-99983 
0-99986 
0-99989 

1-0002 
1-0001 
1-0001 

0-0182 
0-0165 
0-0149 

0-0182 
0-0165 
0-0149 

4-7 
4-8 
4-9 

5-0 

74-20321 

74-20995 

0-99991 

1-0001 

0-0135 

0-0135 

50 

5-1 
52 
53 

82-0079 
90-6334 
100-1659 

82-0140 
90-6389 
100-1709 

0-99993 
0-99993 
0-99994 

1-00007 
1-00007 
1-00006 

0-01219 
0-01103 
0-00998 

0-01219 
0-01103 

0-00998 

5-1 
52 
5-3 

5-4 
5-5 
5-6 

110-7009 
122-3439 
135-2114 

110-7055 
122-3480 
135-2150 

0-99995 
0-99996 
0-99997 

1-00005 
1-00004 
1-00003 

0-00903 
0-00818 
0-00740 

0-00903 
0-00818 
0-00740 

54 
5-5 
5-6 

5-7 
58 
59 

149-4320 
165-1483 
182-5174 

149-4354 
165-1513 
182-5201 

0-99998 
0-99998 
0-99998 

1-00002 
1-00002 
1-00002 

0-00669 
0-00606 
0-00548 

0-00669 
0-00606 
0-00548 

5-7 
5-8 
5-9 

60 

201-7132 

201-7156 

0-99999 

1-00001 

0-00496 

0-00496 

60 

61 
6-2 
6-3 

222-9278 
246-3735 

272-2850 

222-9300 
246-3755 
272-2869 

1- 
1- 

1- 

1- 
1- 
1- 

0-00449 
0-00406 
0-00367 

0-00449 
0-00406 
0-00367 

61 
62 
63 

64 
65 
66 

300-9217 
332-5701 
367-5469 

300-9233 
332-5716 
367-5483 

1- 
1- 
1- 

1- 
1- 

0  00332 
0-00301 
0-00272 

0-00332 
0-00301 
0-00272 

64 
65 
66 

67 
68 
69 

406-2023 
448-9231 
496-1369 

406-2035 
448-9242 
496-1879 

1- 
1- 
1- 

1- 
1- 
1- 

0-00246 
0-00223 
0-00202 

0-00246 
0-00223 
0-00202 

67 
68 
69 

APPENDIX 


309 


TABLE  OF  SINES,  COSINES,  TANGENTS,  COTANGENTS,  SECANTS  AND  COSECANTS 
OF  HYPERBOLIC  ANGLES.— continued. 


It. 

Sinli.  ". 

Cosh.  ". 

Tanh.  u. 

Goth.  u. 

Sech.  a. 

Cosech.  ii. 

it. 

70 

548-3161 

548-3170 

1- 

1- 

0-00182 

0-00182 

70 

7-1 

r>o.V!)S31 

605-9839 

1- 

1- 

0-00165 

0-00165 

71 

72 

(iiill-7150 

669-7158 

1- 

1- 

0-00149 

0-00149 

72 

7-3 

7  u  HUM 

740-1503 

1- 

1- 

0-00135 

0-00135 

73 

7-4 

817-9919 

817-9925 

1- 

1- 

0-00122 

0-00122 

7-4 

75 

904-0209 

904-0215 

1- 

1- 

0-00111 

0-00111 

7-5 

INDEX 


ABBREVIATED    hyperbolic    formulae 
for  current  propagation  in  finite 
lines,  88 
Addition  of  two  complex  quantities, 

10 
Aerial  telephone  lines,   loading  of, 

266 
JEther,  the,  48 

,,      theories,  48 
Alternate     current     potentiometer, 

215 

,,  current  potentiometer  of 

Drysdale-Tinsley,  con- 
nections of,  219 
Alternating  currents,  measurement 

of,  210 
,,  voltages,   measurement 

of,  213 

Amplitude  of  air  motion  in  sound, 
experiments  by  Lord 
Eayleigh  on  the,  91 
,,  ,,    sine  curve,  4 

Analysis     of     complex     curve    by 

Fourier's  theorem,  99 
,,        of  sounds,  Von  Helmholtz's 
experiments  on  the,  102 
Anderson -Bridge,  208 
Anderson  -  Fleming      method      of 

measuring  inductance,  208 
Anglo  -  French    loaded    telephone 
cable,  constants 
of  the,  251,  290 
,,  loaded    telephone 

cable  of  1910... 
279 

loaded    telephone 
cable,    tests    of 
the,  291 
Arrival,  curves  of,  153 


Attenuation  constant  of  Anglo- 
French  loaded  tele- 
phone cable,  288 

,,  constant  of  a  cable 

calculation  of  the, 
245 

,,  constant  of  a  loaded 

cable,  formula  for 
the,  246,  250 

,,  constant  of  a  line,  69, 

256-261 

,,  constant,  measure- 

ment of,  219 

,,  length  of  a  cable,  268 

BARRETTER,  Cohen,  212 

,,  ,,         used          for 

measurement 
of  impedance, 
225 

CABLE,  distortionless,  107 

,,      primary  constants,  practical 

measurement  of,  222 
Cables,  primary  constants  of,  2 

,,       telephonic,  90 
Calculation   of    the   voltage   at  the 
receiving  end  of    a   cable   when 
open,  243 

Calculus  of  complex  quantities,  the,  9 
Campbell,  G.  A.,  127,  129 
Campbell's    theory    of    the    loaded 

cable,  126 
Capacity,  electric,  188 

,,         practical  measurement  of, 

202 

,,         of  cylinder,  191 
,,          ,,  sphere,  188 

,,  submarine  cable,  194 


312 


INDEX 


Capacity  of  a  telegraph  wire,  192 

Chamber  for  loading  coils  on  under- 
ground telephone  circuits,  274 

Clock  diagram,  5 

Cohen,  B.  S.,  210 

„       Barretter,  the,  212 

Complex  quantities,  6 

Concentric  cylinders,  capacity  of,  194 

Constants  and  data  of  cables,  256 — 
262 

Continuously      loaded       submarine 
telephone  cables,  list  of,  278 

Cooper,  W.  E.,  291 

Cremieu,  V.,  52 

Curb  sending  on  cables,  166 

Curl  of  a  vector,  definition  of  the, 
57 

Current  on  a  telephone  line,  pre- 
determination of  the,  233 

Currents,  instantaneous  value  of,  2 

Cur  ve  of  sines,  3 

Curves  of  arrival,  153 

DIFFERENTIAL  equations  expressing 
the  propagation  of 
an  electromagnetic 
disturbance  along 
a  pair  of  wires,  66 
,,  equations  for  propa- 

gation of  electro- 
magnetic disturb- 
ance through  the 
sether,  58 

Distortionless  cable,  107 
Dot  signal,    graphic  representation 

of,  162 
Drysdale,  C.  V.,  214,  215,  216,  217, 

219 
,,         phase  shifting  transformer, 

214 

,,         potentiometer,  216 
Duddell,  W.,  210,  291 
DuddelPs  thermogalvanometer,  211 

EFFECT  of  loading  aerial  lines,  re- 
marks of  H.  V.  Hayes  upon  the, 
269 


Electric    measurements    of    cables, 

necessity  for,  187 
,,         strain,  47,  49 
Electromagnetic  medium,  the,  47 

,,  waves  along  wires, 

59 

Everett,  Prof.,  145 
Example    of    analysis    of    complex 

curve  by  Fourier's  theorem,  100 
Exponential  theorem,  the,  14 

,,  values  of  the  sine  and 

cosine,  12 

FLEMING,  J.  A.,  176,  187,  203 

Formula  for  the  attenuation  con- 
stant of  a  cable,  245 

Formulae  of  hyperbolic  trigo- 
nometry, 27 

Fourier's  theorem,  94 

,,  ,,         proof  of,  97 

Fundamental  constants  of  a  tele- 
phone line,  practical  measurements 
of  the,  231 

GALVANOMETER,  vibration,  218 
Geometric  mean  distance,  199 
German  loaded  aerial  lines,  267 
Gill,  F.,  254.     See  Preface. 
Graphic  representation  of  the  hyper- 
bolic function  of  complex  angles, 
29 

HARMONIC  analysis,  94 
Hayes,  H.  V.,  269,  270,  271 
Heaviside,  Oliver,  106,  108,  133 
Helmholtz,  Von,  102 
High  frequency  currents,  propaga- 
tion of,  along  conductors,  171 
Hyperbola,  area  of  an,  19 

,,          description  of  the,  17 
Hyperbolic  functions,  21 

,,  ,,          curves  repre- 

senting varia- 
tion of,  26 

,,  ,,          inverse,  41 

,,  „          mode  of  calcu- 

lating, 22 


INDEX 


313 


Hyperbolic  functions,  tables  of,  23. 

A  Iso  see  Appendix, 
sector,  23 

,,          sine  and  cosine,  20 
,,          trigonometry,  15 
,,  ,,  formulae 

of,  25 

IMPEDANCE,  final  receiving  end,  85 
,,  ,,     sending  end,  85 

,,  initial  sending  end,  of  a 

line,  72 

,,  of    various     telephonic 

apparatus,     practical 
measurement  of,  222 
Inductance,  formulae  for,  195 

,,  of  parallel  wires,  197 

,,  practical    measurement 

of,  208 
Initial     sending     end     impedance, 

measurement  of,  221 
,,         sending  end  impedance  of 

a  line,  72 

Introductory  ideas,  1 
Inverse  hyperbolic  functions,  41 

JUDD,  W.,  291 

KELVIN,  Lord,  145 

Ivempe,  H.  E.,  187,  215 

Kennelly,    Dr.,    discussion    of    the 

effects     of     leakage     on     loaded 

cables  by,  296 
Kennelly,  A.  E.,  81,  128,  296.      ,SVe 

also  Preface. 
Kingsbury,  J.  E.,  291 
Krarup,  0.  E.,  276 

LAKE   Constance,   loaded  telephone 

cable  laid  in,  279 
Laws  of  reflection  of  electromagnetic 

waves  travelling  along  wires,  65 
Laying  of  the  Anglo-French  loaded 

telephone  cable,  289 
Leakance  on  loaded  telephone  cables, 

292 
Limitations  of  telephony,  104 


Line  integral  of  a  force,  57 
Lines  of  force,  51 
Loaded  aerial  telephone  lines,  266 
,,       aerial    telephone    lines    in 

Germany,  267 
„       cables,  113 
,,       cables,  attenuation  constant 

of,  245 

,,  cables,  effect  of  leakance  on 
the  attenuation  constant 
of,  294 

,,       cables  in  practice,  263 
,,        coils  as  used  in  aerial  lines, 

266 
,,       submarine  telephone  cables, 

274 
,,       submarine  telephone  cables 

in  Denmark,  276 
,,       underground  cables,  271 
Loading  coil  of  National  Telephone 

Company,  273 

,,         coils,  manner  of  inserting 
in     a     telephone     line, 
273 
, ,         coils      of     Anglo  -  French 

telephone  cable,  281 
Loops    and    nodes  of    potential  on 

wires,  175 
Longitudinal  waves,  43 

MAGNETIC  effect  of  a  moving  electric 

charge,  53 
flux,  47,  49 

Martin,  A.  W.,  246,  298 

Maxwell,  J.  Clerk,  200 

Meaning  of  symbol/,  7 

Measurement  of  capacity  of  leaky 
condensers  by  Sumpner's  watt- 
meter, 205 

Medium,  the  electromagnetic,  47 

Model  illustrating  the  mode  of  varia- 
tion of  potential  along  a  long  tele- 
phone line,  73 

Modulus  of  a  complex,  8 

NEUMANN'S  formula  for  inductance, 
197 


314 


INDEX 


O'MEARA,  Major,  247,  275,  280,  286 


TENDER,  H.,  52 

Perry,  J.,  97,  246 

Phase  difference  of  curves,  4 

,,      shifting  transformer  of  Drys- 

dale,  214 
Potentiometer,     Drysdale  -  Tinsley, 

216,  217 

Power  absorption  of  telephonic  in- 
struments, 229 

Practical  measurement   of   capacity 
of  telegraph  and  tele- 
phone cables,  202 
,,         measurements,  187 
Predetermination  of  the  current  at 
any    point    on     a     cable,    under 
simple     harmonic      electromotive 
force,  233 

Product  of  two  complexes,  13 
Production     of     stationary     electric 

oscillations  on  helices,  176 
Propagation  constant,  measurement 

of,  220 
,,  constant  of  a  telephone 

line,  68,  255 

,,  length  of  a  line,  72 

,,  of  air  waves,  43 

,,  current  along  a  line 
short-circuited  at 
the  receiving  end, 
84 

„  ,,  currents     along     an 

infinitely       long 
cable,  71 
,,  ,,  currents  in  telephone 

cables,  71 

,,  ,,  currents    in    a    sub- 

marine     cable, 
theory  of  the,  142 
,,  ,,  electric       currents 

along  leaky  lines, 
182 

„  ,,  electromagnetic 

waves      along 
parallel  wires,  61 


Propagation  of  high  frequency  cur- 
rents along  wires, 
171 

,,  ,,  simple  harmonic  cur- 

rents along  a  finite 
line  with  receiving 
instrument  at  the 
far  end,  86 

,,  ,,  simple  harmonic  cur- 

rents along  a  line 
of      finite     length 
open    at    the     far 
end,  79 
Pupin,  M.I.,  109,  110,  111,  117,  123, 

263 
Pupin's  law  of  loading,  123 

,,       theory  of  the  loaded  cable, 

117 

,,  ,,         of      the      unloaded, 

cable,  110 


QUALITIES    essential  in    telephonic 

speech,  265 
Quotient  of  two  complexes,  13 


EAYLEIGH,  Lord,  91 
Eeed,  0.  J.,  109,  140 

Eeflection  of  electromagnetic  waves 

at  the  ends  of  a  circuit,  63 
Eelation     of      electric     strain    and 

magnetic  flux,  55 
Eepresentation    of    a    vector    by   a 

complex,  8 

,,  ,,     simple    periodic 

quantities    by 
complex 
quantities,  6 
Eoeber,  E.  F.,  133,  141 

,,       theory     of     the     Thompson 

cable,   133 

Eoot-mean- square  value,  3,  6 
E.  M.  S.  value  of  a  curve,  3 
Eotation  of  a  vector,  symbol  for  the, 

11 
Eowland,  H.  A.,  52 


INDEX 


315 


SIGNAL,  telegraphic,  158 

Signals  as  received  on  various  types 

of  submarine  cables,  169 
S- Signal  as  sent  and  received  on  a 

cable,  163 
Sine  curve,  3 

Size  of  a  complex  quantity,  13 
Specification   of    the  Anglo-French 
loaded    telephone    cable    laid    by 
British  Post  Office,  282 
Speed  of    signalling  on   submarine 

cables,  164 
Stationary  oscillations  on  finite  wires, 

174 

Submarine  cable,  capacity  of,  194 
,,          cables,      duplex     trans- 
mission, 168 

,,  ,,        for  long  distance 

telephone     cir- 
cuits.   Paper  by 
Major  O'Meara 
on,  276 
,,  ,,        signals  sent  along, 

various,  169 
,,  ,,       speed  of  signalling 

on,  165 

theory  of ,  142, 146 
,,          telephone  cables,  loading 

of,  274 

Sumpner,  W.  E.,  205,  206 
Syphon  recorder,  157 

TABLE  of  impedances  of  telephonic 

apparatus  (B.  S.  Cohen),  228 
Tables  and  data  for  assisting  calcu- 
lations, 253 
,,       of     hyperbolic    functions    of 

complex  angles,  35 — 40 
Telegraph  wire,  capacity  of,  192 
Telegraphic  signals,  157 
Telephonic  cables,  90 

,,  speech,  effect  of  attenu- 
ation length  of  the 
cable  on,  268 

,,  transmission  measure- 
ments (Coheji  and 
Shepherd),  229 


Telephony,  general  explanation  of, 

90 
,,          practical     improvement 

of,  105 

,,          limitations  of,  104 
Terminal  taper  of  loaded  lines,  269 
Theorem,     useful,     in     hyperbolic 

trigonometry,  86 

Theory    of   propagation    of    simple 

harmonic         currents 

along  a  telephone  line, 

71 

,,          ,,    submarine  cable,  Lord 

Kelvin's,  145 

,,  ,,  the  building  up  of  the 
current  and  potential 
in  a  telephone  line  of 
finite  length,  82 

Thompson   cable,   attenuation    con- 
stant of,  139 
Thompson,  S.  P.,  106,  109,  132,  133, 

139,  140,  263 
,,  ,,     inductively  shunted 

cable  of,  133 
Tinsley,  H.,  157,  169,  170,  215,  216, 

217,  235 

,,        vibration  galvonometer,  218 
Trigonometry,  hyperbolic,  15 

,,  ,,  formulae 

of,  2,3 

UNDERGROUND     telephone     cables, 
loading  of,  271 

VARIOUS    modes    of    expressing    a 

complex  quantity,  12 
Vector  diagram  of  currents  in  a  cable 

(Tinsley),  236 
,,      various  modes  of  representing 

a,  11 
Verification  of  formulae,  233 

, ,  . ,  formula  for  the  ratio 

of  the  currents  at 
sending  and  receiv- 
ing end  of  a  tele- 
phone cable,  237, 238 


316 


INDEX 


Voltage  at  receiving  end  of  a  cable, 

calculation  of  the,  243 
Vowel     sounds,     wave     forms     of, 

92 


WATTMETER,  Sumpner's,  205 
Wave  length,  72 


Wave  length  constant,  measurement 

of,  220 
,,          ,,  ,,         of  a  line,  69, 

256—261 
,,      motion,  43 
Waves,  longitudinal,  43 
Wilson,  H.  A.,  56 
Wood,  E.  W.   52 


BRADBURY,  AGNEW,  &  co.  LD.,  PRINTERS,  LONDON  AND  TONBRIDGK. 


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